1. Introduction. (Received 9 April 2013; accepted 3 June 2013)

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1 1008. Noliear trasiet egagemet characteristics of plaetary gear trai Xueyi Li Shoubo Jiag Sashuai Li Qigliag Zeg NONLINEAR TRANSIENT ENGAGEMENT CHARACTERISTICS OF PLANETARY GEAR TRAIN. Xueyi Li 1 Shoubo Jiag 2 Sashuai Li 3 Qigliag Zeg 4 College of Mechaical & Electroic Egieerig Shadog Uiversity of Sciece & Techology Qigdao Chia 1 lixueyi07@tsighua.org.c 2 jiagshoubo@126.com 3 lisashuai@126.com 4 qlzeg@163.com (Received 9 April 2013; accepted 3 Jue 2013) Abstract. Based o the egagemet priciple of the gear drive the oliear trasiet dyamic characteristics of the plaetary gearig are researched ad the correspodig dyamic equilibrium equatios for su gear plaet gear ad iteral gear are respectively derived. The trasiet egagemet simulatio aalysis of the plaetary gear trai is performed i ANSYS. The correspodig stress-time curves of su gear plaet gear ad iteral gear are obtaied ad the worst egagig locatio ad the ultimate stress i every gear are calculated accurately. The simulatio experimet shows that the egagemet characteristics of the plaetary gearig at ay time ad i ay locatio ca be accurately aalyzed by trasiet egagemet aalysis. It provides a reliable guaraty for the subsequet fatigue aalysis ad structural optimizatio of the plaetary gear trai. Keywords: plaetary gear trai dyamic characteristics cotact stress bedig stress. 1. Itroductio Plaetary gear trai has the advatages of compact structure large trasmissio torque high efficiecy ad stable trasmissio. It is oe of the basic structures of the gear reducers ad is widely used i various mechaical settigs. Durig the process of high-speed rotatio of the plaetary gear trai the primary failure forms of the gear are tooth fracture ad tooth surface fatigue pittig or gluig which are respectively caused by the cotiuous actios of the tooth root bedig stress ad tooth surface cotact stress. For this reaso tooth root bedig stress ad tooth surface cotact stress are the two basic criteria for desigig plaetary gear trai. Therefore it is of great sigificace to make a thorough study of the variatio rules of the bedig stress ad cotact stress throughout the whole egagemet cycle. May scholars have coducted related research o the stregth of the plaetary gear trai. Wag et al. [1] performed the vibratioal aalysis of the plaetary gear trais by fiite elemet method. Meghdad et al. [2] used Dempster-Shafer theory to moitor the vibratio coditio of the plaetary gear trai. Li [3] ad Vecchiato [4] respectively simulated the cotact performace of plaetary gear trai by fiite elemet aalysis software. Through simulatio aalysis the distributio of the cotact stress o each gear is obtaied ad the chagig tred of tooth surface cotact stress accordig to the pressure agle ad radius of the plaetary gear is summarized. Satoshi [5] ad Ishida [6] aalyzed the effect of the load ad backlash o the bedig stress of each gear i a plaetary gear trai ad pipoited the locatio of the maximum stress i each gear. Sudararaja [7] calculated the cotact stress of the su gear ad plaetary gear respectively by fiite elemet method. Duog [8] ivestigated the iflueces of some gear structure parameters such as fillet radius flage thickess pressure agle ad helix agle o the tooth root bedig stress of the gear i a plaetary gear trai. Ko K. [9] revealed that it is very importat to calculate the tooth root bedig stress of the su gear accurately for desigig plaetary gear trai. However all the above-metioed documets did ot fully cosider the trasiet egagemet characteristics of the plaetary gear trai. I fact both the tooth surface cotact stress ad tooth root bedig stress are chagig oliearly i a egagemet period. It is highly ecessary to study the oliear trasiet egagemet characteristics of plaetary gear trai. The purpose of this study is to propose a oliear trasiet dyamic aalysis method for revealig the dyamic behaviors of plaetary gear trai. First the oliear trasiet egagemet VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE VOLUME 15 ISSUE 2. ISSN

2 characteristics of the tooth surface cotact stress ad tooth root bedig stress i the process of plaetary gear trasmissio are cocretely aalyzed based o the priciple of gear egagemet. The the trasiet dyamic equatios of the su gear plaetary gear ad gear rig are respectively derived. Next simulatio aalysis is performed for illustratio. Fially some typical dyamic characteristics of the plaetary gear trai are summarized. 2. Noliear egagemet characteristics of plaetary gear trai 2.1. Noliear trasiet characteristics of the tooth surface cotact stress Accordig to the Hertz cotact theory [10] the tooth surface cotact stress betwee a pair of gears ca be calculated by the followig equatio: F J ( 1 ± 1 ) ρ σ J = 1 ρ 2 πl ( 1 μ μ 2 2) E 1 E 2 (1) where F J is the total meshig force actig o the tooth surface L is the legth of the cotact lie ρ 1 ad ρ 2 are respectively the radiuses of curvature of the two gears at the egagig positio E 1 ad E 2 are respectively the elastic modulus of the two gears μ 1 ad μ 2 are respectively the Poisso s ratios of the two gears. The plus ad mius sigs are respectively used to show the gear pair is of exteral egagemet or of iteral egagemet. I the process of plaetary gear trasmissio for ay gear pair such as su gear ad plaetary gear gear rig ad plaetary gear L chages at ay time F J ρ 1 ad ρ 2 also exhibit oliear behavior. From equatio (1) we ca see that σ J varies oliearly with the cotiuous chage of egagemet poit Noliear trasiet characteristics of the tooth root bedig stress Fig. 1 shows the bedig load actig o the tooth. As show i Fig. 1 poit A is the egagemet poit of the gear tooth. F N is the load actig o the gear tooth which ca be decomposed ito a circumferetial force F t ad a radial force F r. h is the distace from the egagemet poit A to dagerous sectio BC γ is the agle betwee F N ad F t s is the width of the dagerous sectio BC. Y A F t P i N ti γ F ti T t h F r F N N t1 F t1 O X P 1 F t2 B C s Fig. 1. Bedig load actig o the tooth N t2 P 2 Fig. 2. Loads actig o su gear 934 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE VOLUME 15 ISSUE 2. ISSN

3 Based o the catilever beam theory i mechaics of materials [11] the tooth root bedig stress at the dagerous sectio ca be defied by the followig equatio: σ k = 6F khcosγ bs 2 (2) where σ N is the tooth root bedig stress b is the tooth width. Durig the egagemet process of plaetary gear h ad γ chage cotiuously alog with the rotatio of plaetary gear trai while F N also oliearly chages. Equatio (2) shows that σ N exhibits a oliear trasiet behavior throughout the trasmissio of the plaetary gear trai. 3. Noliear trasiet dyamic aalysis of plaetary gear trai 3.1. Trasiet dyamic equilibrium equatio of the su gear Loads actig o the su gear are show i Fig. 2. I order to facilitate the aalysis a auxiliary coordiate system XOY is firstly established. The origi of the coordiate system O coicides with the ceter of the gear. P 1 P 2 ad P i (i = ) are respectively the first secod ad ith egagemet poits betwee the su gear ad plaetary gears. F t1 F t2 ad F ti are respectively the frictio forces at the egagemet poits. N t1 N t2 ad N ti are respectively the egagig forces at each egagemet poit. T t is the torque actig o the su gear. Accordig to Fig. 2 the trasiet dyamic equatio of the su gear ca be writte as: [m t ]δ + [c t ]δ + [k t ]δ = N ti (t) + F ti (t) (3) where [m t ] is the mass matrix of the su gear [c t ] is the dampig matrix while [k t ] is the stiffess matrix δ δ ad δ are respectively the vectors of displacemets speeds ad acceleratios of all the odes i the su gear N ti (t) is the summatio of egagig forces at momet t F ti (t) is the summatio of frictio forces at momet t. At ay momet the directio of N ti is perpedicular to the taget lie of the tooth profile at the poit P i ad the value of N ti ca be calculated by the followig equatio: N ti (t) = T ti L ti T ti = T t { (4) where T ti is the torque actig o the poit P i L ti is the distace betwee the poits P i ad O. I a plaetary gear trai plaetary gears are evely distributed. Hece the torques actig o the su gear at differet egagemet poits are equal. The they ca be calculated by: T ti = T t. (5) As for the frictio force F ti (t) at ay egagemet poit P i its directio is parallel to the taget lie of the tooth profile ad its value is defied by: VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE VOLUME 15 ISSUE 2. ISSN

4 F ti = fλ it t L ti (6) where f is the frictio coefficiet λ i is the directio coefficiet of the frictio force at P i Trasiet dyamic equilibrium equatio of the plaetary gear Fig. 3 shows the forces actig o the plaetary gear. Poits Q 1 ad Q 2 are respectively the egagemet positios with su gear ad gear rig. F x1 ad F x2 are respectively the frictio forces at the egagemet poits. N x1 ad N x2 are the egagig forces at the poits Q 1 ad Q 2. Y N x2 Q 2 F x2 N x1 F x1 O X Q 1 x1 Fig. 3. Forces actig o the plaetary gear N cj Y G j F cj T c N c1 O X G 1 F c1 F c2 N c2 G 2 Fig. 4. Loads actig o the gear rig 936 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE VOLUME 15 ISSUE 2. ISSN

5 By establishig a auxiliary coordiate system as metioed above the trasiet dyamic equilibrium equatio of the plaetary gear is represeted as: [m x ]u + [c x ]u + [k x ]u = N x1 (t) + N x2 (t) + F x1 (t) + F x2 (t) (7) where [m x ] is the mass matrix of the plaetary gear [c x ] is the correspodig dampig matrix while [k x ] is the stiffess matrix u u ad u are respectively the vectors of displacemets speeds ad acceleratios N x1 (t) N x2 (t) ad F x1 (t) F x2 (t) are respectively the egagig forces ad frictio forces at Q 1 ad Q 2. N x1 (t) F x1 (t) are respectively the reactive forces of the egagig forces ad frictio forces actig o the su gear at the same egagemet poit. Hece their values are the same. The values of the N x2 (t) ad F x2 (t) ca respectively be calculated by equatios (8) ad (9): N x2 (t) = T c L c F x2 (t) = fη ct c L c (8) (9) where T c is the torque actig o the gear rig L c is the distace betwee the poits Q 2 ad O η c is the directio coefficiet of the frictio force at Q Trasiet dyamic equilibrium equatio of the gear rig The loads actig o the gear rig are show i Fig. 4. Here G 1 G 2 ad G j (j = ) are respectively the egagemet poits with correspodig plaetary gear. As itroduced above the trasiet dyamic equilibrium equatio of the gear rig ca be writte as: [m c ]ω + [c c ]ω + [k c ]ω = N cj (t) + F cj (t) (10) j=1 j=1 where [m c ] is the mass matrix of the gear rig [c c ] is the dampig matrix [k c ] is the stiffess matrix ω ω ad ω are respectively the vectors of displacemets speeds ad acceleratios N cj (t) is the summatio of egagig forces F cj (t) is the summatio of frictio j=1 forces. N cj (t) F cj (t) are respectively the reactive forces of the egagig forces ad frictio forces actig o the correspodig plaetary gear at the same egagemet poit. Hece their values are the same ad ca be calculated by the equatios (8) ad (9). The displacemets of the egagemet poits i su gear plaetary gear ad gear rig at ay time t ca be obtaied by the equatios (3)-(10). The the correspodig strais ca be obtaied by usig geometry equatios ad stresses ca be calculated by usig costitutive equatios of the materials [12]. Equatios (3) (7) ad (10) show that the trasiet characteristics of the plaetary gear trai are directly related to may factors such as structure material ad loads ad the displacemets strais ad stresses exhibit asychroous oliear trasiet variatio. I traditioal static aalysis method slidig frictio ad iertial force have ot bee cosidered. At the same time the egagig characteristics of the su gear plaetary gear ad gear rig have bee aalyzed at the same egagemet positio. Hece the results ca t accord with the fact. To accurately obtai the egagig characteristics of the plaetary gear trai it is essetial to perform systematic oliear trasiet dyamic aalysis as described above. j=1 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE VOLUME 15 ISSUE 2. ISSN

6 4. Fiite elemet simulatio example 4.1. Trasiet dyamic simulatio of the plaetary gear trai To verify the above coclusios simulatio aalysis is applied by the trasiet dyamic aalysis techique of the ANSYS software. To improve efficiecy a simplified model of the plaetary gear trai is built based o the followig priciples: (1) Select parts of the tooth of the su gear plaetary gear ad gear rig to build the FEM model. The tooth umber of each gear must be accurately determied by the gear ratios to esure that at least oe tooth i each gear should pass through a complete egagemet process durig the simulatio. (2) As the stress of the gear tooth is the primary factor which causes the gear failure the hub ad spoke of the gear ca be simplified to a rigid rig. Table 1 shows the structure parameters of a plaetary gear trai with fixed axes. The iput power is 310 KW the agular velocity of the su gear is 95 r/mi. Table 1. Structure parameters of the plaetary gear trai Structure parameters Su gear Plaetary gear Gear rig Tooth umber Z Module m/mm 14 Pressure agle α/( ) 20 Modificatio coefficiet δ Tooth width d/(mm) Ceter distace a /(mm) 263 Accordig to the method itroduced above a simplified FEM model of the plaetary gear trai is built as show i Fig. 5. The model is meshed by both the mappig ad sweepig modes. As the mesh desity of the gear tooth is of great importace to the simulatio results all the fiite elemet meshes of the teeth are refied to improve the simulatio precisio. To apply speed ad torque two ier stiffeig regios are respectively formed by the ier surfaces of the su gear ad the plaetary gear ad their ceters of rotatio. Similarly a outer stiffeig regio is formed by the outer surface of the gear rig ad its ceter of rotatio. Fig. 5. Stiffeig FEM model of plaetary gear trai 938 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE VOLUME 15 ISSUE 2. ISSN

7 Fig. 6. Cotact pairs of plaetary gear trai To simulate the trasmissio of the plaetary gear system cotact pairs betwee plaetary gear ad both the gear rig ad the su gear are respectively created as show i Fig. 6. Durig the simulatio process speed ad torque are respectively applied to the rotary ceter of the iput ad output gears the they are trasferred to the whole gear through the correspodig stiffeig regios. At the ceter of each gear a revolutio joit is costructed to simulate the rotatio of the gear. The other freedoms of the gears are full costraied. A clockwise drivig speed is applied to the su gear while the aticlockwise workig torque is applied to the gear rig. I order to esure the covergece speed ad torque are respectively applied firstly i a slope curve ad the i steady value. Selectig the augmeted Lagrage method as the cotact algorithm for solvig the cotact problems the trasiet stresses strais ad displacemets of the plaetary gear trai throughout the whole egagemet process ca be obtaied Aalysis of simulatio results For ay ode i the tooth surface of the gear its stress varies with the chage of egagemet positio. Hece it is essetial to perform time-history aalysis to extract the maximum stress i every egagemet positio. The the maximum stress ad the correspodig worst egagemet positio throughout the complete egagemet process ca be determied. Normally the materials of the su gear plaetary gear ad gear rig are differet ad there is o clear chage rule of cotact stresses betwee two sets of egagemet positios where the plaetary gear respectively egages with the su gear ad the gear rig. Hece two time-history curves of cotact stress are costructed as show i Fig. 7. Fig. 7. Time-history curves of cotact stresses From Fig. 7 we ca see that there are two critical meshig positios with ultimate cotact stresses. The first oe appears to be the worst cotact positio betwee the su gear ad the VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE VOLUME 15 ISSUE 2. ISSN

last pair of teeth is leavig the egagemet regio (Fig. 9). The correspodig maximum cotact stresses i the two worst positios are respectively 517.435 MPa ad 437.445 MPa. Fig. 8.

8 plaetary gear where their cotact state trasfers from multi-tooth cotact to sigle tooth cotact (Fig. 8). The secod meshig positio with maximum cotact stress appears to be the worst cotact positio betwee the gear rig ad the plaetary gear where the ext pair of teeth is eterig the egagemet regio ad the last pair of teeth is leavig the egagemet regio (Fig. 9). The correspodig maximum cotact stresses i the two worst positios are respectively MPa ad MPa. Fig. 8. The first worst cotact positio Fig. 9. The secod worst cotact positio The root stress of the gear o the tesile side is usually the basis for calculatig the bedig fatigue stregth of gear. For each gear i a plaetary gear trai there exists a positio where the root stress achieves its maximum value. Fig. 10 shows the three time-history curves of root stresses of the su gear the plaetary gear ad the gear rig. The bedig stress of each gear i ay time ca be extracted from the figure. Fig. 10. Time-history curves of root stresses The bedig fatigue life of the gear is determied primarily by the maximum root stress durig the whole egagemet period. Hece determiatio of the maximum root stress ad the correspodig egagemet positio is the critical cotet of the dyamic aalysis of the gear. As show i Fig. 10 the maximum root stresses of the su gear ad the plaetary gear are respectively MPa ad MPa ad the correspodig egagemet positios are respectively the highest positio of the sigle tooth egagemet of the su gear ad of the plaetary gear. The maximum root stress of the gear rig is MPa. Similarly it is also located at the worst egagemet positio of the gear rig. For the gear rig its overlap ratio is always more tha two its worst egagemet positio lies at the boudary positio where the gear rig trasfers from three-tooth egagemet to two-tooth egagemet. 940 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE VOLUME 15 ISSUE 2. ISSN

9 5. Coclusios The oliear dyamic characters of the plaetary gear trai were aalyzed ad simulated. Based o the ivestigatios the followig coclusios ca be offered: (1) Durig the egagemet process of plaetary gear trai both the cotact stress of the tooth surface ad root stress of the tooth root exhibit oliear trasiet behavior. (2) There are two worst egagemet positios where the cotact stresses of the two gear pairs reach the maximum values. Meawhile there are three worst egagemet positios where the root stresses of the three gears respectively attai the maximum values. (3) The time-history curves of the cotact stresses betwee two cotact pairs are idepedet ad asychroous. Similarly the time-history curves of the root stresses of su gear plaetary gear ad gear rig are also asychroous. I order to esure the stregth requiremet of the plaetary gear trai it is ecessary to check the stregth of the plaetary gear trai at all the five dagerous egagemet positios. Ackowledgemets This work was supported by the Project for Scietific Developmet Pla of Shadog Provice (No. 2011GGX10320) Doctoral Fud of Miistry of Educatio of P. R. Chia (No ) Shadog Provicial Natural Sciece Foudatio of Chia (No. ZR2010EM013). Refereces [1] Wag P. Cai X. Vibratioal aalysis of plaetary gear trais by fiite elemet method. Joural of Vibroegieerig Vol. 14 Issue p [2] Meghdad K. Hojat A. Mahmoud O. Ashka M. Vibratio coditio moitorig of plaetary gears based o decisio level data fusio usig Dempster-Shafer theory of evidece. Joural of Vibroegieerig Vol. 14 Issue p [3] Li C. Itegratio of fiite elemet aalysis ad optimum desig o gear systems. Fiite Elemets i Aalysis ad Desig Vol. 38 Issue p [4] Vecchiato D. Tooth cotact aalysis of a misaliged isostatic plaetary gear trai. Mechaism ad Machie Theory Vol. 41 Issue p [5] Satoshi O. Root stresses ad bedig fatigue breakage of plaet gear. Bulleti of the JSME Vol. 27 Issue p [6] Ishida T. Bedig stress aalysis of idle gear with thi rim. Bulleti of the JSME Vol. 28 Issue p [7] Sudararaja S. Ami S. Fiite elemet aalysis of rig gear casig splie cotact. Joural of Propulsio ad Power Vol. 7 Issue p [8] Duog L. Evaluatio of rig gear tooth stress. America Society of Mechaical Egieers Vol. 4 Issue p [9] Ko K. E. Lim D. H. Kim P. Y. Park J. A study o the bedig stress of the hollow su gear i a plaetary gear trai. Joural of Mechaical Sciece ad Techology Vol. 24 Issue p [10] Igacio G. P. Jose L. I. Alfoso F. Implemetatio of Hertz theory ad validatio of a fiite elemet model for stress aalysis of gear drives with localized bearig cotact. Mechaism ad Machie Theory Vol. 46 Issue p [11] Hassa A. R. Thaigaiyarasu G. Ramamurti V. Effects of atural frequecy ad rotatioal speed o dyamic stress i spur gear. Egieerig ad Techology Vol. 48 Issue p [12] Hasapour K. Ziaei Rad S. Mahzoo M. A large deformatio framework for compressible viscoelastic materials: costitutive equatios ad fiite elemet implemetatio. Iteratioal Joural of Plasticity Vol. 25 Issue p VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE VOLUME 15 ISSUE 2. ISSN

Damped Vibration of a Non-prismatic Beam with a Rotational Spring

Damped Vibration of a Non-prismatic Beam with a Rotational Spring Vibratios i Physical Systems Vol.6 (0) Damped Vibratio of a No-prismatic Beam with a Rotatioal Sprig Wojciech SOCHACK stitute of Mechaics ad Fudametals of Machiery Desig Uiversity of Techology, Czestochowa,