Kunming, , R.P. China. Kunming, , R.P. China. *Corresponding author: Jianing He

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1 Applied Mechanics and Mateials Online: ISSN: , Vol. 540, pp doi: / Tans Tech Publications, Switzeland Reseach on Involute Gea Undecutting with Mechanical Mechanics Based on Neual Netwo Xulei Deng 1,a, Jun Xiao 1,2,b, Jianing He 1,c, *, Wuxing Ma 1,d, Yan li 1, Jinsong li 1 1 Faculty of Mechanical and Electical Engineeing, Kunming Univesity of Science and Technology, Kunming, , R.P. China 2 Depatment of Tanspotation Engineeing, Kunming Vocational and Technical College of Industy, Kunming, , R.P. China a dlhdxl@163.com, b xiaojun @126.com, c hjning@126.com, d mawuxing@126.com *Coesponding autho: Jianing He Keywods: Neual netwo; Involute gea; Undecut amount; MATLAB simulation Abstact. This pape is based on the neual netwo nonlinea mapping chaacteistics. The involute gea tooth numbe, modulus and thei elationship model of cutting the amount effect ae established espectively. Though MATLAB simulation gets the elation cuve between the numbe of teeth and modulus. These cuves can intuitive eflect the tooth numbe and modulus effects on cutting amount change, which povides valuable theoetical efeence fo the design of the involute gea. Intoduction The neual netwo is submitted based on the existing eseach esults in the field of neuoscience, which is a ind of abstaction, simplification and simulation. It can eflect some of the basic featues of human bain function. It is compised of a lage numbe of simple pocessing elements (neuons, pocessing elements, electonic components, etc.) which ae connected with each othe. It is a nonlinea dynamic system of adaptive and self-leaning [1]. Among them, the neuon is the basic pocessing unit of the neual netwo. Geneally speaing, it is a multi input single output nonlinea dynamic system. Pocessing infomation is detemined by the inteaction between neuon stuctues to achieve. Neual netwo computing is a new computing system by simulating the bain paallel infomation pocessing system. It has some geneal nonlinea dynamical systems common: non balance, unpedictability, extensive association and adaptability. The theoetical calculation about undecutting value In the equation of involute in ectangula coodinates. Mae the gea cente at the oigin point, the tooth thicness cente line of symmety coincidence with the longitudinal coodinate Y axis. Thus, get the oot adius δ and cutting amountδ. If at tooth pofile any point adius vecto size is, the tooth pofile angle isϕ. The paamete equation of any point on the tooth pofile is: xi = i δ sinϕi δ (1) yi = i δ cosϕi δ When the numbe of teeth is z, module is m. The standad involute gea of efeence cicle pessue angle isα, which any point of the tooth pofile angle isϕ. Among them: ϕ = invα + π / (2 z) invθ (2) In type 2: θ is an involute spead angle at an abitay point. As shown in figue 1. When the line NK along a cicle as the pue olling, the locus MK is any point on the line K, which is the involute of All ights eseved. No pat of contents of this pape may be epoduced o tansmitted in any fom o by any means without the witten pemission of Tans Tech Publications, (ID: , Pennsylvania State Univesity, Univesity Pa, USA-03/03/16,23:59:22)

2 Applied Mechanics and Mateials Vol the cicle. This line NK is called the place line of involute, angleθ is called the flae angle of point K. is the adial adius at any point K of involute MK [2]. When this involute as gea tooth pofile, and mesh with its conjugate tooth pofile in point K. Fig.1 The geneating of involute Acute angle between the tooth pofile in the nomal diection of the point and the velocity diection is called the point of pessue angle. The tooth pofiles in the nomal diection of the point and the velocity of the acute angle between α called the point of pessue angle. Fom the gaph, we can analyze to the type (3): tan α = NK MN = b (3) b Accoding to Figue 1 geomety, can be futhe deived type (4): tanα = α + θ (4) By fomula (3) visible: flae angle θ is function of pessue angleα, called the involute function. Synthesis of abitay point pessue angle α and (3), we can get the pola coodinate equation of involute, as shown in fomula (5): α = accos b / θ = tanα α Integated type (1) and (2), we can get the paametic equation of involute: (5) cosα sin[ invα + π / (2 z) invθ ] xi = 2 cosθ (6) cosα cos[ invα + π / (2 z) invθ ] yi = 2 cosθ Gea tooth oot tansition cuve is fomed in cutte tooth top ound in the geneating pocess [3][4]. Gea tansition cuve is the fomation of the tool cente point of fillet, which is the equidistant cuve of extending involute. So the paametic equation of the tooth oot tansition cuve is:

3 94 Advanced Reseach on Mechanics, Manufactuing Engineeing and Applied Technology II 2( α cot α + b) α 2( α cot α + b) x = sin cos 2 + α sinα (7) 2( α cot α + b) α 2( α cot α + b) y = cos cos α 2 + sinα The simultaneous equations (5), (7), we can deive the involute gea undecut amount adius δ : δ = ( cos α ) / 2 cosθ (8) If the involute gea occu undecutting phenomenon, the cutting amount δ should be equal to the diffeent between oot adius δ and the adius of the base cicle gea b. Theefoe, we get the fomula of cutting amountδ : cosα 1 δ = ( 1) (9) 2 cosθ Analysis of type (9) is not difficult to find: thee is a elationship between undecut amountδ, modulus m and the tooth numbe z, flae angleθ. By type (9), we can calculate: when the involute gea undecutting, the specific value of the undecut. But also can analysis the elationship between undecut amountδ, modulus m and the numbe of teeth z. But due to the two sets of paamete equations (5) and (7) ae all complex function of expansion angle, which can not solve by geneal method. If use the nonlinea mapping chaacteistics of neual netwos. Accoding to the elationship between the gea modulus and teeth numbe and amount of undecut to simulate the modulus and undecut elation cuve. It can diectly to undecut the numeical expession [5]. It has geat significance fo gea design to eliminate the defects in the tansmission pocess. Simulation and analysis about standad involute spu gea undecutting The modulus of standad involute gea of m=5 as example. Selection of a hidden laye netwo, node numbes of hidden laye select 5. The netwo stuctue is 1 * 5 * 1 single hidden laye fowad netwo net=newff(inputn,outputn,5). In the netwo taining, leaning pocess shows the fequency Plot Inteval is set to 1epochs. The biggest taining step fo epoch=1000 times, eo squae and index select mm. Respectively simulate modulus of m=5 involute gea in diffeent systems using BP neual netwo and undecut amountδ cuve unde diffeent numbe of teeth. The taining paametes fo TainLM: 8/20000 epochs, mu=le+10. The sum of eo squae SSE= As shown in Figue 2, is the undecutting cuve based on MATLAB fo standad involute spu gea [6]. Fom the figue 2 we can seen: when the numbe of teeth is equal to 17 tooth fom the taditional theoy, undecut amount is not 0. But when the numbe of teeth z is equal to 22, the cutting amountδ is equal to 0. When the numbe of teeth is less than 22, undecut cuve above the base cicle. So the oot cicle adius is lage than the adius of base cicle, thus occu undecutting phenomenon. And the diffeence of the two adiuses is the undecut amountδ. When the numbe of teeth is geate than o equal to 22 teeth, the oot cicle and cicle cuve coincide, and thus will not poduce the undecut phenomenon. Accoding to the above figue analysis, we can see: in the standad involute spu gea pocessing, we must conside the addendum tool pat c*m. Fo the involute standad gea, not occu undecut teeth numbe is 22. Instead of at home and aboad, most of the cuent elated wos and in the liteatue mentioned: based on the theoy of taditional undecut, put the ac cutte machining gea and gea meshing with a tal in ode to get the 17 tooth.

4 Applied Mechanics and Mateials Vol Fig.2 Undecutting value chaacteistics gaph of standad involute gea with modulus m = 5 Summay This pape intoduces the neual netwo technology, discusses the neual netwo model and the leaning pocess. In MATLAB envionment analysis the standad involute gea undecutting value, modulus and numbe of teeth elationship. Apply neual netwo nonlinea chaacteistics in the numbe of teeth o modulus of has been set envionment conducted a moe accuate simulation to involute geas undecut amount. Accoding to the amount of undecut simulation cuve, we can intuitively now the numbe of teeth and the modulus effect on the amount of undecut. The undecut value is deceased with inceasing numbe of teeth. Though the above eseach and analysis, we not only solved the long-standing mistaen ideas in the taditional cutting theoy, and edefined the minimum teeth numbe of not occu undecutting about standad involute gea. Refeence [1] Litvin F.L. Development of Gea Technology and Theoy of Geaing, NASA Refeence Publication,1998. [2] R.Maiti, A.K.Roy. Minimum tooth diffeence in intenal-extenal involute gea pai. Mech.Mach.Theoy,1996, 31(4): [3] Peng Nanhua, Chen Qiangen, He Jilin. The analysis of undecut gea model based on Matlab[J]. Manufactuing Infomation Engineeing of China, 2007,36(9): [4] Zeng Zhouliang, Chen Zhigang. Analysis on the least geas in gea mechanism[j]. Machiney Design and Manufactue, 2005, 11, [5] Satish Chand. An appoach to the detemination of spu gea tooth oot fillet. Jounal of Mechanical Design, 2004, 126: [6] Hui G. The Minimum Numbe Teeth of Involute Gea Design and Undecut Contolling[J]. Shandong Jiaotong Keji, 2006(3):77-80.

5 Advanced Reseach on Mechanics, Manufactuing Engineeing and Applied Technology II / Reseach on Involute Gea Undecutting with Mechanical Mechanics Based on Neual Netwo /