Basic Law of the Flat Interlocking of Involute Cylindrical Gears with Asymmetric Profiles

Transcription

1 Inernaional Journal of All Research Eucaion an Scienific Mehos (IJARESM) ISSN: 455-6, Volume 5, Issue, January 07, Impac Facor:.87 Basic Law of he Fla Inerlocking of Involue Cylinrical Gears wih Asymmeric Profiles Genai Tsveanov, Sveoslav Simeonov, Anreas Pournaras 3,,3 Deparmen "Mechanical an Precision Engineering" Technical Universiy - Gabrovo, Bulgaria ABSTRACT This aricle is eucion of he necessary an sufficien coniion for geomeric synhesis of olue cylinrical gears wih asymmeric profile of he eeh by he classic approach. The necessary coniion is presene for slack-free engagemen of he corresponing olues along he iniial circles an he crossing of he lines of engagemen along he iner-cener sraigh line, which is a new inerpreaion of he Willis's basic law of he fla engagemen. Base on he heorem for reversing he irecion of movemen, o consequences are rawn, allowing generaion of asymmeric ooh profile along various polois an achieving qualiaive inicaors of engagemen, which canno be achieve wih a symmeric profile. Keywors: basic law of he fla inerlocking, possibiliies of generaion, olue cylinrical gears, asymmeric profile INTRODUCTION The classic heory of gearing akes as inpus he parameers of he ool neee for proucion of he oohe wheels of he gear. For he geomeric synhesis of an olue cylinrical gear, four oupu parameers are necessary: moulus of he gear - m, profile angle of he oupu conour -, coefficien of isplacemen of he oupu conour - x, he number of eeh of he wheels of he gear - z. These four parameers eermine he axiomaic esign of gear mechanisms uner cerain parameers of he ool. The use of asymmeric olue cylinrical gears eliminaes he uncerainy of he selecion of inepenen parameers an a "free geomeric synhesis" can be implemene. SUMMARY Axiom of he ool grip for asymmeric profile of he eeh The uniformiy in he proucion of oohe wheels by he meho of rolling is provie by inroucing he so calle "oupu conour" which is sanarize [,,3,4]. The oupu conour is a recilinear rack, whose sies are incline a profile angle =0. The line along which he ooh hickness is equal o he wih of he iner ooh space (s=e=p/) is calle pich line. The so calle moifie oupu conour also exiss. The geomerical calculaion of he olue gears is one on he basis of he sanar elemens of he conour. The oupu generaing conour fills he iner ooh space of he oupu conour similar o shape an mol. The oupu generaing conour serves for profiling he eeh of he ooh-cuing ools. The ooh of he generaing conour is ivie ino o by he pich line - hea an base. If he heigh of he hea is equal o he heigh of he base, he generaing conour is calle equally high an if, a he pich line he hickness of he ooh is equal o he hickness of he iner ooh space - evenly ivie (uniform). The generaing conour is he orhogonal projecion of he cuing eges of he ooh cuing ool - rack ype on he fron surface of he cu oohe wheel. I is he base of he ooh-cuing ool, however i oes no accoun for he fron an back angles, as well as oher inheren characerisics of he ool. 6

2 Inernaional Journal of All Research Eucaion an Scienific Mehos (IJARESM) ISSN: 455-6, Volume 5, Issue, January 07, Impac Facor:.87 In he engineering, an equally high conour is use - mill wih open profile, while in precision engineering - mill wih full profile is use. Therefore, he cuing eges of he ool eermine he shape an he nominal imensions of he olue profiles of he ooh when wrapping he work-piece. This saemen efines he ool for proucion of wheels using cenroial wrapping, o which one shif correspons relaive o he pich circle. Therefore, he axiom of ool gearing for an asymmerical profile is ransforme in he form of: The generaion of eeh wih asymmerical profile of olue cylinrical gears allows changing he moule, he profile angles of he oupu conour an he coefficien of shifing in such proporions ha he poloi circles of he gear wheels remain consan.. For beer clariy in he inerpreaion of he geomerical parameers of he gears, he following clarifying conceps are aope - primary an seconary axis of symmery of he oupu conour. "Primary axis of symmery" is he axis iviing he ooh of he oupu conour ino o pars wih equal hicknesses. The asymmerical ooh profile acually oes no have such characerisics, hese are aope for he use of a cerain analogy beeen he exising relaionships in he lieraure, base on he esablishe ones. In he correlaions of he symmeric profile, he "+" sign refers o olue cylinrical wheels wih exernal eeh, while he "-" sign refers o olue cylinrical wheels wih inernal eeh. Coefficien of isplacemen of he wheel wih inernal eeh means he isplacemen of he profile of he ool of he equivalen wheel wih exernal eeh. Apar from he axiom of insrumenal gearing, anoher hypohesis se forh in he classical heory is he one of "congruence of he generaing pairs. Accoring o his hypohesis, congruen generaing pairs are he pairs, whose generaing surfaces can be brough ino such a posiion ha hey coincie wih each oher, imposing one anoher a all poins.. From his hypohesis, i follows ha he o wheels of he gearing are manufacure wih one ype of ool, while he asymmeric profiles of he oohe wheel are generae by he iner ooh spaces of he ool uring he relaive movemen of he work-piece relaive o he ool. Fig.. Oupu (OC) an generaing conour (GC) of an asymmeric oohe profile in fron secion Fig. shows an oupu conour of an asymmeric profile in he fron secion, which iffers from he sanar ones, by: The presence of o ifferen profile angles, an equal pich, boh beeen he corresponing an beeen he opposie profiles. The oohe wheels generae his way are convenionally enoe as I; 7

3 Inernaional Journal of All Research Eucaion an Scienific Mehos (IJARESM) ISSN: 455-6, Volume 5, Issue, January 07, Impac Facor:.87 he presence of equal pich, bu only beeen he corresponing ooh profiles. The gears generae his way are convenionally enoe as II. For his generaion, he following necessary coniion mus be observe: m m. m, () where m, are he fron moules of he oupu conour wih non-uniform hickness, corresponing o profile angle α, for wheels an of he gear (Fig.). m is he fron moule of he oupu conour wih non-uniform hickness, corresponing o profile angle α for wheels an of he gear (Fig.). By using he axiom for insrumenal engagemen an he hypohesis for congruency of he generaing pairs, he following ifferences are foun in he esign of olue cylinrical gears wih asymmeric profile an he eermining of he hickness of he ooh along he pich circle. The eerminaion of he hickness of he ooh of he oohe wheel on he pich circle, when he conour is isplace, is: S, I possibiliy:.m x,.m. g g. () S S II possibiliy:. m m 4. m m 4 x.m. g g ; x.m. g g ; m m. m. (3) Theorem for reversing he irecion of he movemen in case of asymmery of he ooh profile Afer fining he basic geomeric parameers of he ool, geomeric synhesis of he olue gear wih asymmeric profile of eeh follows. Le's hypoheically suppose ha he gear wih asymmeric eeh profile is generae an compleely eermine by o symmerical gears. This is foun easily, using he following relaions for eerminaion of he angles of engagemen ( an ):. x x.g. x x z z ; z.g z, b, ;, b,, (4) where α an α are he profile angles of he ool, proucing he wheels by he meho of cenroi wrapping of he fron secion;, iameers of he iniial cyliners in he fron secion; b, iameers of he main cyliners in he fron secion. 8

4 Inernaional Journal of All Research Eucaion an Scienific Mehos (IJARESM) ISSN: 455-6, Volume 5, Issue, January 07, Impac Facor:.87 Afer efining he angles of engaging an he iner axis isances for boh inepenen symmerical spees, hese shoul also be eermine for he asymmeric gear. If he correlaions for heir eerminaion coincie, i follows ha such gear consiss of o symmerical ones an is an compleely eermine by hem. In orer ha he gear wih asymmeric profile of he eeh is working, a slack-free engagemen beeen he opposie profiles mus be realize along he iniial cyliners: S S,, z z, (5) where S, are he hicknesses of he asymmeric eeh in he fron secion, for wheels an of he gear, corresponing o profile angle, along he iniial cyliners; S, - are he hicknesses of he asymmeric eeh in he fron secion, for wheels an of he gear, corresponing o profile angle, along he iniial cyliners. The examining of he asymmeric ooh as compose of o symmerical ones allows eermining heir hicknesses (S, an S, ) along he iniial cyliners in he fron secion of he wheels: I possibiliy: S.x,.g. r.z z ; (6),,,,,.x.g, S,. r,.z z,,, ; (7) II possibiliy: S.x,.g. r.z z ; (8),,,,, S.m.x,.g,,.r,,.m.z z,,, (9) aking ino accoun he non-proporional ivision of he pich of he oupu conour in he fron secion: m m. m, (0) From (6) o (0), i follows ha he full hickness of he asymmeric ooh for he ifferen possibiliies for synhesis of a cyliner wih a ranom raius is: 9

5 Inernaional Journal of All Research Eucaion an Scienific Mehos (IJARESM) ISSN: 455-6, Volume 5, Issue, January 07, Impac Facor:.87 I possibiliy: II possibiliy: S i,.ri,..z, x,. g z, i, g i, ; () S i.r i S i.r i m m x. g g.. 4.m.z i i m m x. g g.. 4.m.z i i m m. m. z z ;, () The heorem of Willis is accepe as vali, ensuring consancy of he gear raio for boh pars of he profile when he irecion is reverse: i, cons, (3) while he expression (5) akes he following form:. x x z z.g. x z x z.g. (4) Correlaion (4) expresses he general form of he requiremen for slack-free engagemen in case of asymmeric ooh profile realize a ifferen profile angles an compose of o inepenen symmerical profile of he wheel. Acually, he wheels of he gear wih asymmeric profile have one cenroi because only one ool generaes he ooh profiles. This requiremen generaes he following aiional equaion: 0

6 Inernaional Journal of All Research Eucaion an Scienific Mehos (IJARESM) ISSN: 455-6, Volume 5, Issue, January 07, Impac Facor:.87 b b. (5) This proves ha he iniially accepe assumpion for o iniial cyliners of each wheel of he gear is false. Then correlaion (4) akes he following final form for olue cylinrical gears: x x.. g g. (6) z z Therefore, he eerminaion of he engagemen angles of he asymmerical ooh profile is one in accorance wih he following sysem: b b. x x. g g z z. (7) The ranscenenal sysem (7) iffers from he originally accepe sysem (4), which suggess ha he asymmeric ooh profile exis inepenenly ousie he symmeric profiles ha consruc i. When he profile angles of he ool of he sysem (7) are equal, he correlaion is obaine for eermining he angle of engagemen for a symmerical profile (4). Anoher way of obaining he sysem (7) is by subsequen summaion of he lef an righ pars of he correlaions for a symmerical profile (4). The given sequence for eucion of correlaions from (4) o (7) is proof of he following Theorem: Reversible olue cylinrical gears wih asymmeric profile of he eeh (an consan gear raio) is realize when he iner axis isance is equal, eermine for boh profiles, an in case of slack-free engagemen of he corresponing profiles along heir iniial circles. The heorem for reversing he irecion of he moion gives he necessary slack-free engagemen of he profiles an sufficien coniion equal iner axis isance, allowing synhesis of such gear. For he firs ime he heorem for reversing he irecion of movemen has been efine by auhors eam of he Technical Universiy - Gabrovo in 005 6,7, an o ocor's egrees have been efene an one habiliaion hesis has been evelope. Due o he long research in he fiel of geomeric synhesis, he heorem for reversing he irecion of movemen is now inerpree as a basic law of he plane gearing for reverse olue cylinrical gears, while he following consequences have been summarize. Consequences of he heorem for reversing he irecion of movemen Consequence I: Equal hicknesses of he eeh of he joinly working wheels of he gear along he poloi circle are obaine wihin one complee pich angle (.π/z) Possibiliy I for generaion. This approach assumes ha all variables in he ranscenenal sysem (7) are known, while i is solve in relaion o he unknown angles of engagemen an : ( n ) ( n ) ( n ) g ( n ) arc g arc.. ( n ) ( n ).

7 Inernaional Journal of All Research Eucaion an Scienific Mehos (IJARESM) ISSN: 455-6, Volume 5, Issue, January 07, Impac Facor:.87. x x.( g g ) / z z g ( n ) g arc. ( n ) (8) Characerisic for his possibiliy of generaion is ha he sufficien coniion for equaliy of he iner axis isance is reuce o he following form:. (9) From correlaion (9), i follows ha he possibiliy for generaion has he following isavanages: he profile angles of he ool α an α mus be known in avance; when a large ifference beeen he profile angles is use, he heigh of he ool is ecrease; he full form of he ranscenenal sysem (7) is no use; here is a wie range of aa abou gears wih symmeric profile, which canno be exrapolae irecly ono he asymmeric profile. All hese isavanages are eliminae by using a new opion in he synhesis, which is eermine by he following Consequence II: The synhesis of gears wih asymmeric profile allows opimizing of he engagemen of gears wih symmeric profile. Fig. Difference beeen he iniial symmery axis OO' an he new axis OO'' for Possibiliy II for generaion in he fron secion of he wheel Base on consequence II of he heorem for reversing he irecion of movemen, o new approaches are offere for solving he ranscenenal sysem (7). For he geomerical inerpreaion of hese o possibiliies i is assume ha he full from of he ranscenenal sysem (7) is implemene, which implies ha he asymmery beeen he profiles is expresse by he basic circles an no only by he ifference beeen he profile angles of he ool. This

8 Inernaional Journal of All Research Eucaion an Scienific Mehos (IJARESM) ISSN: 455-6, Volume 5, Issue, January 07, Impac Facor:.87 implies a ifference in he "iniially" accepe axis ОО. Fig. shows he ifference beeen he iniial symmery axis OO' an he "new" axis OO'' in he fron secion of he wheel. Hypoheically, we will assume ha we can se he posiion of his new axis ОО. I is assume ha he profile angles of he oupu conour α an α in he normal cross-secion are known (as in possibiliy I). Base on he ranscenenal sysem (7) he engagemen angle α is eermine by he meho of successive approximaions 5 unil he esire accuracy: ( n ) ( n ). z z. x x. g g g ( n ) ( n ) g ( n ). (0) The profile angle of he ool (α ) in he fron secion is eermine by he sufficien coniion of he heorem an correlaion (5) relaive o he iniial axis. The general form of ha oher sequence of solving he sysem is:. x z x.g z b. b x x. g g z z. () Apar from his varian of soluion, he ranscenenal sysem (7) offers anoher possibiliy for he same iniial assumpions, bu in a ifferen sequence. The engagemen angle α is eermine base on he raio of he main circles, which is known in avance: b arc. () b The profile angle of he ool α is eermine epening on he correlaion on slack-free gearing (6) as a epenen variable unil he esire accuracy: ( n ) ( n ). z z. x x. z z x x. g g x x. z z.. ( n ) ( n ) ( n ).g ( n ) ( n ).g ( n ). (3) 3

9 Gear Angles of he conour for opimize profile Thicknesses of he eeh along he crown circles Overlap raio Specific slipping Inernaional Journal of All Research Eucaion an Scienific Mehos (IJARESM) ISSN: 455-6, Volume 5, Issue, January 07, Impac Facor:.87 The engagemen angle α relaive o axis ОО is eermine by he sufficien coniion of he heorem an correlaion (5).. x b b z x.g z. x x. g g z z. (4) Such opimizaion of he asymmeric profile base on he heorem for reversing he irecion of movemen has been implemene for an oupu oluion cylinrical gear wih ouer engagemen an he following parameers of he oupu conour: m=mm, α=α=0, h a =, h f =,5, isplacemen of he conour x =x =0,5, number of eeh of he wheel of he gear - z =0, u=.5, β=0 an open profile milling meho. The aa abou he main qualiaive inicaors an comparison wih he oupu symmeric profile are given in Table. As a basic parameer for he opimizaion (Table ) he crierion of maximum fron overlap raio ε α =max was use. One opimal varian has been eermine wihou clipping a ineger value of α. This possibiliy of solving obains increase of he fron overlap raio ε α =,33 for symmeric profile up o ε α =,44 for asymmeric profile possibiliy II. Table Qualiaive inicaors of symmeric an asymmeric profiles generae hrough Possibiliy II Symmeric α = 0 Sa = 0,6 Sa = 0.70 ε α = ε α =,33 θ a =, θ e =,7 Asymmeric II α = 5 Sa = 0.67 Sa = 0.76 ε α =,44 θ a =,05 θ e =,74 This propose new possibiliy of generaion has he following avanages: uses he available aa for a symmeric profile; allows synhesis of asymmeric profile, where he slack-free engagemen is achieve relaive o an axis ifferen from he iniial one; he limiaion of he ool is eliminae, while he profile angle α varies wihin he accepe proporionaliy beeen he o basic (generaing) circles. From correlaion () an (4), i follows ha possibiliy II for synhesis oes no change he necessary an sufficien coniion for reversing he irecion of movemen, accoring o he heorem for reversing he irecion. Therefore, he olues generae hrough possibiliy I are equiisan o hose generae hrough possibiliy II. 4

10 Inernaional Journal of All Research Eucaion an Scienific Mehos (IJARESM) ISSN: 455-6, Volume 5, Issue, January 07, Impac Facor:.87 Compare o possibiliy I for synhesis, possibiliy II offers anoher sequence for solving he ranscenenal sysem (7), an namely a conversion of he classic heory of ooh engagemen an creaion of new gears hihero unknown qualiaive an srengh parameers. CONCLUSIONS. Base on he classical heory of ooh engagemen, he following are efine: axiom of he ool engagemen in case of asymmery of he ooh profile, basic parameers of he oupu conour, necessary for he proucion of wheels by he meho of cenroi wrapping an he iniial geomeric coniions for synhesis of conjugae wheels an gears.. A heorem has been efine an proven abou he reversing of he irecion of movemen of olue cylinrical gears wih asymmeric profile of he eeh, hrough which heir geomeric synhesis is possible. 3. The exisence of o coniions was proven for reversing he irecion of he olue cylinrical oohe gears wih asymmeric profile, proviing slack-free engagemen an equaliy of he iner axis isance. 4. Two consequences have been efine of he heorem for reversing he irecion of movemen, allowing ifferen saring polois of he gear wheels, as well as various improve qualiaive inicaors of he ooh engagemen, which i is impossible if he eeh have symmerical profile. 5. The exisence was proven of o possibiliies for generaion of he asymmeric ooh profile, giving ifferen qualiaive an srengh parameers a equal iniial variables of he synhesis, which has been shown by example of opimizaion of a gear wih asymmeric profile of he eeh accoring o he crierion of maximum fron overlap raio ε α =max. 6. The axiom for ool engagemen of olue cylinrical gears wih asymmeric profile of he eeh, he heorem for reversing he irecion of movemen an he resuling consequences are he basic law on gearing of a new generaion of oohe gears by using he classic approach of geomeric synhesis wih parially explore o his momen inicaors of qualiy, srengh an precision. REFERENCES []. Arnauov K. e. al., Machine elemens, Sofia, "Technique", 980; []. Takov I., e. al., Free geomeric synhesis, olue cylinrical gears Geomery, Sofia, "Technique", 98. [3]. Sabchev P., Mealworking ools, Sofia, "Technique", 98. [4]. Tays B.A., Accuracy an conrol gears, Moscow, Mechanical engineering, 97. [5]. Bronsein I.N. Semenyaev KA, Hanbook of mahemaics for engineers an suens of Technical Schools, Moscow, "Science", 986. [6]. S. Simeonov, G. Tsveanov, Theorem reversing he irecion of he olue cylinrical gears wih asymmeric profile of eeh, Proceeings of he Technical Universiy of Gabrovo, IM - "Vasil Aprilov", 3, Gabrovo, 005, ISSN [7]. S. Simeonov, G. Tsveanov, Theorem reversing he irecion of he olue cylinrical gears wih asymmeric profile of eeh, Mechanics of machines, 63, Varna, , ISSN