Vol. 4, No. (9) 34-4, DOI:.4874/ti.9.4..5 Triology in Industry www.triology.rs RESEARCH A Metod for Predicting Contact Strengt and Life of Arcimedes and Involute Worm Gears, Considering te Effect of Wear and Teet Correction M. Cernets a a Faculty of Mecanical Engineering, Lulin University of Tecnology, ul. Nadystrzycka, 36, -68 Lulin, Poland. Keywords: Worm gear Mesing of Arcimedes and involute gears Teet correction of te worm weel, Contact strengt Wear Gear life Corresponding autor: Myron Cernets Faculty of Mecanical Engineering, Lulin University of Tecnology, ul. Nadystrzycka, 36, Lulin -68, Poland. E-mail: m.czerniec@pollu.pl A B S T R A C T A new metod developed y te autor is used to investigate te effect of wear of worm weel teet on variations in teir curvature radius and tat of teet correction in Arcimedes and involute worm gears on teir contact strengt, wear and life. Te effect of teet correction on tese variales is determined, for ot constant and canged contact conditions. It as een found tat wen a positive correction coefficient is applied, te maimum contact pressures and weel teet wear decrease wile te gear life increases; wen te correction coefficient is negative, te trend is opposite. On canging contact conditions after wear limit of te teet, te contact pressures are consideraly lower, te gear life is practically te same, and te wear of worm weel teet is sligtly iger tan tat in te case of constant conditions. To accelerate calculations, te lock-cumulative metod developed y te autor and not te specified step-y-cumulative metod are used. Te computational time is reduced proportionately to te size of te lock of te numer of interactions. 9 Pulised y Faculty of Engineering. INTRODUCTION Worm gears are widely used in macine design ecause tey ensure considerale canges in torque and revolutions frequency wit small dimensions. Wen gear elements are in mes, sliding friction occurs, causing wear of te worm weel teet. It is vital to e ale to predict gear life and wear of worm weel teet already te stage of gear design. Despite te ovious need for tat, te metods for predicting arasive wear of worm gear teet given in te literature [-3] cannot e applied to uncorrected and corrected gears, especially at oundary lurication. Based on elastoydrodynamic lurication teories, te studies [,] report te results of prediction of teet wear in an uncorrected gear y a modified Arcard law, using a model allowing for variations in contact pressures and oil film tickness in contact area. However, it sould e noted tat te Arcards law of arasive wear used in te aovementioned works, assuming linear dependence on slip velocity and contact pressures, in te 34
case of elastoydrodynamic and also oundary lurication does not work. Based on te autor generalized metod for prediction of wear at sliding friction (dry, mied, oundary) [4], were te mecanism of fatigue wear is assumed, wic as een confirmed y te works of te autor as well as in literature y oter researcers, te autor developed a metod [5] enaling prediction of contact parameters, worm weel teet wear and gear life. Tis paper reports te results otained wit te proposed metod wit respect to determination contact and triological variales descriing Arcimedes and involute worm gears, considering te effect of teet wear and correction on tese parameters.. MATEMATICAL MODEL OF WEAR AT SLIDING FRICTION In worm gears, at oil lurication, sliding friction occurs etween te worm coils and te worm weel teet at torque transfer. Assuming tat te wear rate of te elements of te triosystem (see - te left side of equation ()) is functionally related to te level of specific friction force (rigt side of te equation), te kinetics of wear in a sliding triological system is descried y a system of linear integer equations [4]: dk v dt k ( ), k ;. () Eperimental values of te function approimated y te following formula: m were s.35r. k are k k C k sk / () mk Te eperimental values of functions are determined as follows [4]: were L vt. i L/, (3) Te unit friction force is descried y te Amontons-Coulom law of friction: i i fp (4) Te maimum contact pressured p p is ma determined wit te Hertz formula, depending on te numer of mesing pairs w of te worm weel teet: w p ma.564 N/ w (5) were E / / E.. Linear wear of te worm weel teet ( p ma const ) By integration of equation (), considering function () and formulas (4), (5) and assuming tat te contact pressures const, we get a formula for calculating unit linear wear of gear teet during a single time of triocontact interaction: were t / v. w t fp ma (6) m ( w) C s According to te Hertz formula: ( w) Te sliding rate is: Accordingly, m.56 Θ N / w ( ) ( ) v v v (7) (8) cos A were tg A mz /, n /3. e, /u (9) pa It sould e noted tat te dominant effect on te resulting slip speed is velocity v. Te mesing force N is determined y te formula: were: T N d cos sin( ) p 3 T 955 ( N / n ) and arctg f /cos. () Te Arcimedes worm aout te trapezoidal profile (Fig. ) is: 35
te involute worm is: d ( sin epa ) () () Te curvature radius of involute worm tread profiles and worm weel teet. r tg c 3 cos ptg cos c r sin p epa epa r sin p pa e (3) Te worm weel teet ave involute profiles (Fig. ). (were 5 ), f.m (were 5 ); tan mz / d, d qm ; r.5 d, m a a a (were 5 ), a n n m (were 5 ); r.5 zm, r.5 d, ; z uz, q z r e, r.5 d, m q. pa sinp Oter geometrical parameters: - Arcimedes worm gear: 8 mz p arctan( tan ), tan, - Involute worm gear: r.5d cos, tanc tan n/sin, c p tan r c arctan, r arctantan c r, mz d cos, 8 r. r Fig.. Scematic design of a worm gear. Te coordinate is in te range of Accordingly r. m and. A f A B r a. Te section of mesing [ A, B ] must e proportionately divided into smaller sections: A A,, 3 3,... and B n B. 3. GEAR GEOMETRY Te geometry of te analysed worm gears (Fig. ) is descried y te following relationsips: r.5 d,.m f f f B 4. LINEAR WEAR OF TEETH AND LIFE OF THE WORM GEAR ( p ma const ) Te wear of te worm weel teet witin one our of gear operation is calculated in te following way: 6n, n n / u (4) Te life of te worm gear for te acceptale wear of te worm weel teet is calculated according to te formula: t / (5) 5. LINEAR WEAR OF THE WORM WHEEL TEETH ( p ma var ) In tis case, te formula (6) as te form: 36
t w fp ma n m C s m (6) As a result, te radius of curvature of te teet increases wit every revolution of te worm weel, and te maimum triocontact pressures decreases wit increasing te widt of te contact area. Accordingly, p ma.564 N wθρ,.56 θnρ w (7) were = - Arcimedes worm gear, t / v. - involute worm gear, 5. Definition variations in teet curvature radius due to wear Te proposal of a metod is to determine te effect of linear wear of gear teet on te curvature radius. Te variale curvature radii of teet profiles at -t point of contact are epressed: n n ρ ρ λ (8) Terefore, for numerical resolution, a step-ystep calculation of te following parameters is performed: n, ρ, ρ, p ma,, t. To accelerate te calculations, it is epedient to use te lock-cumulative metod developed y te autor and not te specified step-ycumulative metod. Tis metod is ased on an assumption tat, during a given numer of weel revolutions n (interaction lock B), te parameters, p ma,, t are constant. A susequent lock allows for a cange in tese parameters due to wear, and so te calculations are made for new values. Te computational time is reduced proportionately to te size of te lock of te numer of interactions. For eample, in te given solution (item 8) wen te lock size is B = 6 n / u =3377 revolutions, ten tis time will e 3377 times smaller tan in te case of te step-y-cumulative metod. Hence, according to te lock-cumulative metod variale curvature radii: B ρ ρ λ (9) n 6. TOTAL WEAR AND LIFE OF THE WORM GEAR ( p ma var ) Te total wear of worm weel teet determined y te aove step-y-cumulative and lockcumulative metods is: n n, n n were B., () n B Te gear life after revolutions n of a worm weel wen acceptale wear is reaced: t n / 6n. () 7. WORM WHEEL TEETH CORRECTION In worm gears, we can only apply correction to te worm weel teet. Te interaial distance is: were a ( d d)/,. ak a m () Te reference diameter of te worm in an uncorrected gear: dw d m (3) Terefore, te distance e pa etween te -t point of contact and te point of contact will e: e pa rw sin p Oter geometrical parameters are determined in accordance wit te formulas for uncorrected worm gears. 8. NUMERICAL SOLUTION Te computations were performed using te following set of data: N = 3.5 kw, n = 4 rpm, m = 6 mm, z =, u = 5.5, f =.5, q = 8; worm -. 37
ardened steel grade 45 (HRC 5) descried y E =. 5 MPa, μ =.3; worm weel ring ronze CuSn6Zn6P6 descried y E =. 5 MPa, μ =.34; C = 7.6 6, m =.88; τ s = 75 MPa; for = ; ; 3; 4 and 5, respectively = 8; ; ; 4 and 6 mm; * =.5 mm; λ = ; B = 6 n / u = 3377 revolutions ( ours of work); wit doule-pair mesing. Te numerical results are given in Figs. 7: а) Arcimedes worm gear, ) involute worm gear. Solid lines denote constant conditions of contact, roken lines mark variations in te reduced curvature radius due to wear. 5 radius ρz ρ at teir acceptale wear * =.5 mm wen is positive and it decrease at negative values (Fig. 5). As a result of te applied teet correction, wen, and z increase and te gear life t min increases too (Fig. 3). Wen, and z decrease, wic leads to reduced gear life. Taking into account te teet wear, te gear life will e sligtly sorter tan tat in te case wit constant conditions of interaction. t min, 5 5 p ma, MPa 5 5 - -.5.5 p ma, MPa 5 - -.5.5 4 35 3 5 5 5 = =5 =. =5. - -.5.5 = =5 =. ) Fig.. Wear and correction versus wen = ; p wen = 5. ma =5. ma p : p ma Figure sows te relationsips etween te maimum contact pressures p ma and te displacement coefficient on entering te mes ( = ) and on leaving te mes ( = 5). It was found tat te positive correction of te worm weel teet leads to reduced pressures, wile te negative correction results in teir increase compared to te oservations made regarding te uncorrected gear. Tis results in an increase in te teet curvature summary t min, 5 5 5 tmin (.3) tmin (.5) tmin (.3) tmin (.5) - -.5.5 Fig. 3. Wear and correction tmin (.3) versus gear tmin (.5) life min =.3 mm; tmin (.3) tmin (.5) min t min wen * ) t : t wen * =.5 mm. Te linear wear of gear teet during one our of operation, wit contact conditions maintained constant (friction pat ), will e lower tan under variale contact conditions (friction pat ) due to wear (Fig. 4). Wen positive correction coefficients are applied, te wear is lower (Fig. 4) tan tat of te uncorrected gear. As a result of teet wear, te curvature radii of te weel teet (Arcimedes worm) and te reduced curvature radii ρz ρ (involute worm) significantly increase, wen compared to teir initial values (Fig. 5). 38
,mm 3.7E-5 3.E-5.7E-5.E-5 In ot gear types, te sliding rate does not cange at -t contact points, in eiter constant or variale conditions of interaction (Fig. 6). 4 3.5,mm.7E-5 3.7E-5 3.E-5.7E-5.E-5.7E-5 - -.5.5 ħma ħmin ħma ħmin - -.5.5 Fig. 4. Weel teet wear ħma per our versus ħmin cange in ħma ħmin : wen = ; min wen = 5. 6 ma ) V, m/s 3.5 3 4 5 Fig. 6. Variations Ряд in sliding rate Ряд over toot Ряд3 eigt. Ряд4 Ряд5 Te effect of gear teet correction on te contact area widt is insignificant wen pma, const; te value of is affected more significantly wen pma, var due to wear of te worm weel teet (Fig. 7). Te contact area widt consideraly increases due to wear of te gear teet..7.6 p, mm p, mm 8 4 - -.5.5 6 8 4 Pma Pmin Pma Pmin, mm,mm.5.4.3. 3 4 5.7.6.5.4.3 = = = = =- =- - -.5.5 Pma ) Pmin Fig. 5. Wear and Pma correction versus Pmin curvature radii cange: ma ( ma ) wen = ; min ( min ) wen = 5.. 3 4 5 ) = = = = =- =- Fig. 7. Wear and correction versus contact area widt during one interaction etween toot and worm coil: = ; = ; =. 39
Te proposed metod can e used to estalis qualitative and quantitative relationsips wit respect to te effect of teet correction and wear on te load capacity, life, wear, sliding velocity and contact area widt of te worm gear. 9. CONCLUSION. It as een found tat, in contrast to uncorrected gears, te maimum contact pressures decrease wen te correction coefficient is positive. Wen tis coefficient is negative te pressures increase (Fig. ).. Te life of te worm gear increases at ut decreases at (Fig. 3). 3. Te wear of te worm weel teet decreases at and increases at, in contrast to te case wen (Fig. 4). 4. Te toot curvature radii increase wit increasing х and tey decrease wit decreasing (Fig. 5). 5. Gear correction does not ave a significant impact on te contact area widt. 6. Negative correction leads to decreased load capacity and life of te worm gear and increased wear of worm weel teet, ence it is not recommended. REFERENCES [] K.J. Sarif, H.P. Evans, R.W. Snidle, D. Barnett, I.M. Egorov, Effect of elasto-ydrodynamic film tickness on a wear model for worm gears, Proceedings Institutions Mecanical Enginers, Part J: Journal Engineering Triology, vol., pp. 95 36, 6, doi:.43/3565jet [] K.J. Sarif, H.P. Evans, R.W. Snidle, Prediction of te wear pattern in worm gears, Wear, vol. 6, iss. 5-6, pp. 666 673, 6, doi:.6/.wear.6..8 [3] H.G. Sainiak, Wear and life of te worm gears. Pulising House of Lodz University of Tecnology: Lodz, 7. [4] M.V. Cernets, J. Kelinski, R.Ja. Jarema, Generalized metod for te evaluation of cylindrical involute gears, Materials Science, no. 47, pp. 45 5,, doi:.7/s3--9366-9 [5] M.V. Cernets, R.Ja. Jarema, Prediction of te life of te worm gears in Arcimedes and involute worm gears, Prolems of Triology, no., pp. 5,. NOMENCLATURE a is te interaial distance in an uncorrected gear, is te worm weel widt, С k,mk are te indicators of wear resistance of triological pair materials in select wear conditions, d is te reference diameter of te worm, d is te reference diameter of te worm weel, e pa is te distance of point from te contact point, f is te sliding friction coefficient, q is te diametric quotient of te worm gear, k is te linear wear of elements of a triological pair, i consumption of samples of triological vapor materials, f is te eigt of worm tread ase, a is te eigt of ead of te worm coil, n is te linear wear of teet during a single interaction, decreased due to canges in, t, p ma, B is te wear of teet during a lock of interaction, * is acceptale wear te teet of te worm weel, i load level, point of contact etween elements of a kinematic pair (worm worm weel toot), k denotes te numer of an element of te triological pair ( worm, worm weel), L te road of friction, m is te aial modulus of mesing, mn mcos is te normal modulus of mesing, n is te numer of revolutions of te worm weel per minute, n is te numer of revolutions of te worm weel wen te acceptale worm weel teet wear * is reaced, 4
N is te mesing force, N is te transmitted power, n is te numer of revolutions of te worm, p pma are te maimum contact pressured determined wit te Hertz formula, depending on te numer of mesing pairs w of te worm weel teet, p ma are te maimum current triocontact pressures. R m is te temporary tensile strengt of materials, r f is te radius of a circle of worm cavity, r a is te radius of a circle of worm coil prongs, r is te radius of a asic circle of worm coils, t is a period of wear, t is te time of contact etween te mesing elements at -t point on te friction pat wit a ( ) lengt equal to te contact area widt w, T is te torque transmitted y te worm, u is te gear ratio, is te sliding rate at -t point of contact etween elements of a kinematic pair, is a sliding rate during worm gear revolution, is a sliding rate at te point of contact etween te worm weel toot and worm coil, w is te numer of mesing pairs etween worm coils and worm weel teet, is te correction coefficient, z is te numer of worm coils, z is te numer of teet in te worm weel, n is te angle of mesing, c is a face pressure angle, c is te face pressure angle at -t point, is te angle of elevation of te screw line of worm coils, is te angle of inclination of flank pitc line on reference diameter of te gear, is an angular coordinate in every pitc (degrees), is te caracteristic function of wear resistance of triological pair materials under select conditions, λ is a non-dimensional coefficient of wear impact, μ k,ek are Poisson s ratio and Young s modulus of material of te worm weel, respectivity, is te reduced curvature radius etween worm coils and worm weel teet at -t point of mesing, ρ z is te reduced radius of curvature of te worm gear, ρ is te reduced radius of curvature of te involute worm gear; is te reduced radius of curvature of te involute worm gear a result of te wear toot; wear of te steel worm is omitted, is te radius of curvature of te Arcimedes worm gear, is a friction angle, is te unit friction force affecting wear rate of materials, sk is te temporary sear strengt of gear material, θ is te Kircoff modulus, is an angular velocity of te worm. 4