A Complete Review of Planetary Gear Dynamics and Vibrations Researches
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1 "#$ *8/ 01) ,6 07$ *8'(") *+,-.,- *2 *29,: 6 6 -, -. / / , -. / / ,; " * 7 424, ;2 =;7 > : / 3 3 FGH; < 4E D 3 A C; B/ 6A / 19 /?@ => 4; <3 :3 / 43 4; :, C;N F../ 43 4O I 4;<3?.: 2 3 / C;N F J8 M;. JK I D 3 C; 5 1 UJV 4;=T S R/ A R, 3 / O I 4;<3 2 A QH >>P 4 /?.: 2 / [/ XK ZH 3. 6/ 49/ Y5 CV/ A I XK F W FH; /. <; 3. 6/ A F ^>P 4 / FH 4;. \ ] XK 4/RH, <; 6.; C. 43 4O I 4;< ;=?.: O I 4;<3:*; *8?2+ A Complete Review of Planetary Gear Dynamics and Vibrations Researches M. Karimi Khoozani M. Poursina A. Pourkamali Anaraki Department of Mechanical Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran Department of Mechanical Engineering, Isfahan University, Isfahan, Iran Department of Mechanical Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran Abstract Planetary gear boxes are widely used in power transmission systems with high torque to weight ratio, high speed reduction in low space and access to deceleration in several stages. The main objective of this research is an overview on the dynamics and vibrations of the planetary gear systems. This research provides a comprehensive study of relevant resources published in various journals for researchers who wish to work on the dynamics of planetary gear systems. For this reason, these topics have been expressed in several sub-sections that cover all the dynamics and vibrations of planetary gear systems. In the end, a summary of the issues discussed and suggestions for further research in this field are expressed. Keywords: Planetary gear systems, Dynamic, Vibration, Dynamic models. >>P 4 / / 43 4O I 4;<3?.: 43 4O I 4;<3 2 A QH 1 UJV 4;=T S R/ A J 3 / / <; 3. [1],92 G504,D *8=(BC3) 247A 34*8G Z5 3; 69A S =3 ^>P 2 ;O I 2 73 d 4;= /. ( 1970 =3 A 43 4;O I 2 4 / :] 2 A 4A?>>P! 6 A :/. 9 A`.: 4; ao Q. 43 4;O I?.: GV A 4H 1 Q... Z5 ]O ` 2 FGH; 4;O I?.: 2 A. 1 4;>.; 61 O = ;;. ;1 1 Q. H /, / 2 A > 14 A H > 10 Q@K ; 3 K F., 43 4;O I 1990 ; / A F. 1 4;> : 2000 : / / / ;> : Y O = / / 3 / ;=3 F/ =3 10 =b 3 ;>.3 3 ci 2 J. 2 5 ^>P F J8 M; * poursina@eng.ui.ac.ir : 95/01/24 :! 95/11/18 :) *!
2 / : F 9 W 6/ 9 4;Z, = 4 / :[9]./ A Q. / J K K Ω Ω Ω ( 1) Ω c3 T w 3 w B/ : / K HD?nV / A/ CO I D 3 ]3 / P HD,O 19 6/ u 73 w u / Ω. 5 => 4]O CO I ]3 / 73 w w 6@../. 5 A/ 4+" =2 G ED ZH O ED 6/ uj8 d = p F / G uj` ;= F. 9 W *M]: F 9Y/ =K F /.; J K Q.../?3P A / <5K P 6H 4;= ud. 73 S3 3 P 6H = ; H ) A O 4;^>P 4 0 6>>P.[165] 3 w [17] 6 - Q Q. 73 P../ [18] 6HJ3/ /. Z5?:] 2003 =3 Y g.[2] 3 /A/ ;O I ]O ` 4;= 4 ;^>P A H :. ;1 ;> F 3 / /.[3]., > F 43 4;O I 2 A 4;O I 2 / S > [2] R, =i 6D /., > -I S> Y [3] R,., / 4 / 7JV 4;= O 4;=3 =b 4;= F.. j 43 4O I 4;<3 4, 7JV 4;4/3 ^>P M; /, / 6 ]O 3 / 4;= 9 2 A =i 6D/ A. <> ]O ` 4/<> P 6H = 9 4;Z, = 3. 3 / Z 9 CV/ F.[4] 6 *8J, G Q. ) 43 O I <3 4ED H; 9 F 9 Z,?8/ (A/ O O I 6A A Q>) ; ^ b A 9 4;Z,.. = = O I 4 9 ;6@ (6A / m 4 *M]: S> = p F.. Qn 0 / ; o / 4 *M]: = ;6@ O I 4;, : 3 / ;= F.. 9 <3 ED :. 4/ <> 9 A 3 / 4A R, 3 /...?.: 2 A 9?8 4;^>P 4 [16] 4+" =2 GA 42M 01)2-3 <3 :b 4; ;w F: Q. A?.: x :b. 0 <; XP F../ x 4,?.:!3 F 4 / <; <; 4 / W.. 73 / ;3 6> O3 A F 9 2H / 6-4;w 3P 4 / / Z 2 4.O O I 1G = A 73 / 3 3 / 43 <3 2 :b = y / ^>P F y > /.[6] ; j so 1G ) 6/ d D G F.. ;O :b w zy/ { > 4 / :3 d D G = / [6] R, 6 - JJP ) / 3 y -I 3 / H3 W FH; /.[9] / - 3 / u / 4 4;J8. > 0 *N2 01)2 4;) Q. 4,?.: / [/ XK H@ F /!3 ]O ` 4;nV1 QK K=9 DB G. 9 W ED 1O I K S> = F F A >>P. 4A, 2 S> ED ; 49/ [7 6] 6 - [5] 6H; ; Q. 73 =../ [8] 63 L,>-DB G / ED 1G K F 9 W / rd ;= FI 6 P / HD P78 d D 4; K 6 P >>P. 4A, 3 ED ; F //. 9 W Y 39 [9] F Q. 73 = H F A../ [11] 6 - [10] /3 *1- G-3 9 W 4:/ 3 4E <3 4ED = F, 3 : 4A, C. ED ; F //.. 73 = p F A >>P 6 A.>, 3.. [14] Q [13] g [12] 6 - / 6 4 / d D G so 1G 4;= J b / 4 / 4:/3 =. / 3 4;O I / 4;<3../ u3 v O I / 4;<3 6 = 358
3 H JD 3 - AO H HP.[12] / 0 W 9 4; : m A. <0 <H: 6A / m <3 2 / ) F g.[22] Q1 / Z <0 6D P 4, U V9,W *8TCD -4 4;< ]O ` 33 QD ^>P FI Y, /../ Aw 43 4O I ]O?8 / 43 4;<3 - [12 5] ]O `?8 / ;<3 ;> >/ /. 9 W H: O I 4;<3 / /1.. 9 W?.:!3 4?8 / 99 ]O ` 4; F.. ;1 43 4O I 4;<3 4, 07. 2; -3!3 C, ) Q. ;../ ` ) / 43 <3 2 ]O `!3 6 - Q / ^>P F Z5 / :] P 6H w ) 2 P w 5 m / 3 5 F 63. ;1!3 6 \d / ) (4 9 Z5 9 4;Z, ) / FH; O ^>P Y ; 2; -I wb/ QK ) A 61>>P -.[21] ) / 3 F 43 4O I 4;<3 4Y A H; = S RA 3 FI S3 S. I 9.. ;O / RA A <-3 ;3, Q/@ ` C3 T O3 A. 4;]O :@.. ;3 6 / / / RA / 5 ;]O F ; [263] R, ;3 6 / RA 4 / JJP 4;= / / RA x ^>P 2 Y / d D G = / 4?8 43 4;O I / RA A ;^>P 1/ R, / 6 =i 4 /.3 9?8 4A3Q R8 43 4;O I / RA.. [ ].3 3 / [31] R, Y J; *,64 X1Y7 61 ;O I 2 / S UJV R/ 3 / P QD 2 6D/ => 4]O ; => 4]O J K.[2] 3. * O I 7, w / / / / R/ 2 u@ d?8 / 4;<3 4 9 FI 6, /.. 6/ ) 4.O O I =K 43 4O I -43 4;O I O 4;O I A O / <; / np 6 F (0 4.O 4;O I u3 6I / 2 P 4;7ƒ 4;<3 AP *8O+?B: QK 4;) CV/ F -4 : 9 j 43 *4>*,6G2,Q2 *8O+-4 4;<3 2 4 / /O/ 4D 4 9= 0 ) QK 4 / ) F y F //. O. 43 4O I 4 y ) A [10] /3 39../ ;<3 F 43 4O I <3 A 1G = 2 QJP 4 / 4 4D 4;) JH, A <; [5] -/ ) >>P S3 / 78 RS- O+-4 K!3 F 4 / 9 b / 2; wb/ ) F. 73 ]O ` ]O 4;<3 4 ; K!3 2 P 3 3 d F / ) / 6A / m u d 2 P QK ) F.; 2; 4 3 4*0, / wo3.. S/ / 6 Q 7?B: <3 nv1 : : / {HD ) Q 4,?B: S> W 3 ; 4;?D 2; wb/ ) 6 /. 9 W 2;?D. 9 W?D 2 A C/ D XP.. 79 D -I 2; wb/ ) 6 / D 2; wb/ ) ; ;O I 2 wb/ ) ; 63.[209]. 73 D -I / 43 4O I <3 2 D 2; 61>>P -.[21] :3 Aw FI F 9 W :/ / O I 4; Aw A. ]O `? }. U8 /. j [19] 6 - ~.FJ/ ^>P D 2; wb/ ) A 61>>P J8 ud :[3] A 3?D 9?x1 2; 3 4;!3 Qi ]O ` 4; A O / ) A 73 / K / 6 2; -3 A ) F 3 6 / 2; -I wb/. F/.3 P { ) G247. O ;w nv1 : / [/ N > x QK / 6 wo3. 3/ 6 6 / 4; :b. V3 <3!3 = ph5 ) A 73 / Q/@ 6A A Q> 4;<3 4 / S> ) F J K <3 2 / ) F <> b/ 6 -../ 73 V3? m O ^>P A. =HD 43 O I 359
4 / / RA u d QJP / O ^>P 6H; g O v 43 O I DH5 2 4O I <3 2 = QJP 6H; g. ;T.[41] 9 4;Z, = A 73 / /} 4A/ / v 3 / 4 J2 *8'(") * -.0 *2 *89,: 34 ^>P F O CV/ 8 :, R/ 3 / /.. - / 43 4;O I 2 UJV 4;A { > G ;< P 6H 4;= JJP 4;= Q. ;A ZH.3 9?8 ;: / 5 4; FGH; P /, R/ v 43 4;O I 2. j = F R, Y ;> F F/../ Y R, F./ [40] R,, C;N 43 <3 A/ 3. V3 K 4 /?B:? } ^>P F Q FH; /. / /} H; B/ 4;D 3 A/ 6 A. 3 T -, A 4; F // W v 43 4;<3 2 A?:] :3 W / 4:/3 4E?B: V3 A/ ]. ED ; 4 / 4A, 6 F 9 3 T? } 6 43 <3 Y Y H; B/ 4;D 3 A/ 6 A...?B: 43 4;<3 4 / P 6H = 2 j.v / / 5 4; Z5 -, CA Y 2 K b S/ 53P8 v 43 4;O I 4.P 6H = JJP V3 0 O I F 9 W *M]: v 43 <3 / <K?B:.<3!3 4 / F }x 6Y 3 / : ,1 [1] Cooley Christopher G., and Robert G. Parker. A review of planetary and epicyclic gear dynamics and vibrations research, Applied Mechanics Reviews, Vol. 66, No. 4, pp. 1-15, [2] Özgüven H. Nevzat, and Donald Russell Houser. Mathematical models used in gear dynamics a review, Journal of sound and vibration Vol. 121, No. 3, pp , [3] Wang Jianjun, Runfang Li and Xianghe Peng, Survey of nonlinear vibration of gear transmission systems, Applied Mechanics Reviews Vol. 56, No. 3, pp , [4] Yang Jianming and Liming Dai, Survey of dynamics of planetary gear trains, International Journal of Materials and Structural Integrity Vol. 1, No. 4, pp , F./ C;?.: F //. M*K :@ 6D P ao ) Z5 4:/ 3 = 2 / 4 9 :@ }x 6-4 S/ QJP / O ^>P Y.[12] 4 / ]. d?8 / 9 4;O I F/ 2 4 / 2 P 19 F 2 P 4 F 6 M*K A.[32] 3/ / 43 4O I <3 / 6. ;^>P A F 4 0 JH,.. [22] g FGH; [33] F *89,: 34 *+,- S- X>, 02,Z2-7 *2 /.. nv 03 / K : 9 Z 6A A Q>?B:./ Qn 6 =K 4A/ nv1 :./ ;O Y 3 T? } Q. 3 / 43 O I DH5 2 P78 QO?.: 4 / K 9 V3 F S3 3 T? } F 9 W v 43 O I 4:/ 3 = <; Q.[9].: 4; 3 / Z0; 9 W 3 T? }.[34] WM 8 6 A A3 B/ 4;D O I <3 = Y Y 0 ^>P 2 -.[35]. Z5 S3 = =H: ` 3 T 4;<3 N >. ;1 B/ D 3 / 43 4;O I ~B 3 / ,) *8'(") -8 *2 *89,:?.: 2 ^@ / 5 4;^>P../ / R, R/ 43 4;O I 2 4 / / 5 4; O 4;^>P 6 ED ; 4 49/ Z5 43 O I DH5 A, - / F.[39-37] y3. 4: wo3. 3P ED ; 1G >,/, = Y ^ b A 6 / 4; :b 4;w P 6H = 9 4;Z, = y / 3/.[37] > 34 4 J2 *8'(") [+4 *2 *89,: v 43 4;O I 2 A, R/ -9 =.. O - : /./ P S3 /} 4A/ / v 43 < ;Z, R, 3 /.[40]. V ?.: 2 A 9?8 4;^>P 4 360
5 H JD 3 - AO H HP [25] Singh Avinash, Epicyclic load sharing map development and validation, Mechanism and Machine Theory Vol. 46, No. 5, pp , [26] Montestruc Alfred N., A Numerical Approach to Calculation of Load Sharing in Planetary Gear Drives, Journal of Mechanical Design Vol. 132, No. 1, pp. 1-4, [27] Bodas Ajit and Ahmet Kahraman, Influence of carrier and gear manufacturing errors on the static load sharing behavior of planetary gear sets, JSME International Journal Series C Mechanical Systems, Machine Elements and Manufacturing Vol. 47, No. 3, pp , [28] Singh Avinash, Application of a system level model to study the planetary load sharing behavior, Journal of Mechanical Design Vol. 127, No. 3, pp , [29] Singh Avinash, Influence of planetary needle bearings on the performance of single and double pinion planetary systems, Journal of Mechanical Design Vol. 129, No. 1, pp , [30] Ligata H., Kahraman A., and Singh A., An experimental study of the influence of manufacturing errors on the planetary gear stresses and planet load sharing.", Journal of Mechanical Design Vol. 130, No. 4, pp. 1-9, [31] Cheon Gill-Jeong. and Robert G. Parker, Influence of manufacturing errors on the dynamic characteristics of planetary gear systems, KSME international journal Vol. 18, No. 4, pp , [32] Parker R. G., A physical explanation for the effectiveness of planet phasing to suppress planetary gear vibration.", Journal of Sound and Vibration Vol. 236, No. 4, pp , [33] Parker R. G., and J. Lin. Mesh phasing relationships in planetary and epicyclic gears, ASME 2003 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, [34] Eritenel Tugan. and Parker R. G., Modal properties of threedimensional helical planetary gears, Journal of Sound and Vibration Vol. 325, No. 1, pp , [35] Cooley Christopher G. and Parker R. G., Vibration properties of high-speed planetary gears with gyroscopic effects, Journal of Vibration and Acoustics Vol. 134, No. 6, pp. 11, [36] Cooley Christopher G., and Parker R. 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"Experimental quantification of the effects of torque on the dynamic behavior and system parameters of planetary gears.", Mechanism and Machine Theory Vol. 74, pp , [40] Sondkar, Prashant, and Kahraman A., A dynamic model of a double-helical planetary gear set.", Mechanism and Machine Theory Vol. 70, pp , [41] Wang S., Zhang and C., Wang F., The analysis of dynamic load coefficients of double- helical planetary gear sets, International Gear Conference, Lyon, [42] Sheng Zhaohua, et al. Modal Analysis of Double-Helical Planetary Gears With Numerical and Analytical Approach, Journal of Dynamic Systems, Measurement, and Control Vol. 137, No. 4, pp. 17, [5] Hidaka Teruaki, Yoshio Terauchi and Makoto Fujii. Analysis of dynamic tooth load on planetary gear, Bulletin of JSME Vol. 23, No. 176, pp , [6] Kahraman, A. 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W. and Kahraman A., Steady state forced response of a mechanical oscillator with combined parametric excitation and clearance type non-linearity, Journal of Sound and Vibration Vol. 185, No. 5, pp , [20] Al-Shyyab A., Non-linear dynamic analysis of a multi-mesh gear trains using multi-term harmonic balance method, PhD Dissertation, The University of Toledo, [21] Sun, Tao. and HaiYan Hu., Nonlinear dynamics of a planetary gear system with multiple clearances, Mechanism and Machine Theory Vol. 38, No. 12, pp , [22] Wang S., Theoretical and experimental investigations on dynamics of spur planetary gear transmissions based on planet phasing theory, PhD Dissertation, Tianjin University, [23] Ligata H., Kahraman A., and Singh A., A closed-form planet load sharing formulation for planetary gear sets using a translational analogy, Journal of Mechanical Design Vol. 131, No. 2, pp. 1-7, [24] Singh Avinash, Load sharing behavior in epicyclic gears: Physical explanation and generalized formulation, Mechanism and Machine Theory Vol. 45, No. 3, pp ,
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