Mechaical Efficiecy of Plaetary Gear Trais: A Estimate Dr. A. Sriath Professor, Dept. of Mechaical Egieerig K L Uiversity, A.P, Idia E-mail: sriath_me@klce.ac.i G. Yedukodalu Assistat Professor, Dept. of Mechaical Egieerig K L Uiversity, A.P, Idia E-mail: yedukodalu@kluiversity.i Dr. A. Jagadeesh Group Director, GD Rugta College Kohka Kurud Road, Bhilai, Chhattisgarh, Idia E-mail: jagadeesh.ae@gmail.com Received: November 20, 2011 Accepted: November 29, 2011 Published: December 31, 2011 doi:10.5539/mer.v11p97 URL: http://dx.doi.org/10.5539/mer.v11p97 Abstract I this paper a easy method is preseted to estimate the mechaical efficiecy of a gear-trai-whe the umber of teeth o gear wheels is specified. Also, the ifluece of the structure is icluded i estimatig the efficiecy. This helps i comparig the gear trais for mechaical efficiecy. Keywords: Gear Trais, Mechaical efficiecy, Graph theory, Trasmissio, Frictio 1. Itroductio Methods for geeratig distict plaetary gear trais with the specified elemets or gear pairs are available (F. Buchsbaum & F. Freudestei, 1970; R. Ravisaker & T. S. Mruthyujaya, 1985; L. W. Tsai, 1989; C. H. Hsu & J. J. Hsu, 1997). Also, some methods are available to estimate umber of the trasmissio efficiecy of gear trais. These methods are mathematically more rigorous ad do ot lead to a quick estimate. I the preset paper a semi-empirical method is proposed to estimate the mechaical efficiecy of a plaetary gear trai. The mechaical efficiecy ot oly depeds upo the size of the gear wheels but also o their relative arragemet i.e., structure. This aspect is also cosidered. This method gives reasoably good results. The structure of a gear trai is best represeted by a graph ad hece the use of graphs is explaied. 2. Method Assumig that there is o power loss at bearigs ad the etire power loss is due to gear pairs oly, the efficiecy of a simple gear trai ca be expressed by the relatio (M.F. Spotts),. f ( N E 1 N N 1 N 2 ) Where, E is the mechaical efficiecy of a gear pair, f is the coefficiet of frictio, N 1 ad N 2 are the umber of teeth o the matig wheels. As per (J. E. Shigley, Miscke, 2003), the eq. (1) is a excellet approximatio. From eq. (1) it is devious that the efficiecy of a gear pair is maximum whe N 1 = N 2, I other words, great variatio i the umber of teeth leads a great reductio i the trasmissio efficiecy. Eq. (1) is useful i estimatig the trasmissio efficiecy of every gear pair i a gear trai. I order to estimate the 1 2 (1) Published by Caadia Ceter of Sciece ad Educatio 97
trasmissio efficiecy of a gear trai as a whole, i which the gear pairs are arraged differetly i.e structure of a trai, the followig approach that uses graph theory is proposed. 3. Graph Represetatio of Gear-Trais A graph cosists of vertices ad edges. A vertex is represeted by a small circle while a edge is represeted by a lie joiig the vertices. A elemet of a gear trai such as a gear wheel or a carrier is represeted by a vertex of a graph while the joits betwee various elemets of a gear trai area represeted by edges. I gear trais, we have two types of pairs (i) turig pairs ad (ii) gear pairs. I order to differetiate the turig ad gear pairs, turig pair edges are represeted by a sigle lie edge while a gear pair is represeted by a double lie edge. For example the gear trais show schematically i figure.1 (a) is represeted by the graph, figure. 1(b), N 2 ad N 3, the umber of teeth are show i both the figures 1 (a & b). The efficiecy E ij of each gear pair (i - j) with specified umber of teeth ca be estimated usig eq.(1) ad the same ca be show ear the gear pair, figure 1 example, E 23, 4. Applicatio to Gear Trais The simple gear trai figure.1 has three elemets or three vertices. Next group of gear trais cosist of four vertices or two gear pairs. There are oly two distict gear trais with four elemets. They ad their graphs are show i figures 2 ad 3. Kowig the teeth umbers of gear wheels the efficiecy of each gear pair (2-3) ad (2-4) i case of figure. 2 ad gear pairs (1-3) ad (2-4) i figure. 3 ca be determied usig eq. (1). The resultig efficiecies E 23, E 24 etc. are show o the graphs, figures 2 & 3. Figure 2. Gear Trai cosistig of 4 elemets-first Type Figure 3. Gear Trai cosistig of 4 elemets-secod Type 5. Efficiecy Estimate Efficiecy of a system with compoets havig efficiecy E 1, E 2 etc. will deped upo the structural arragemets i.e. whether the compoets are (i) i series (ii) i parallel or (iii) mixed. The estimate of the system efficiecy is explaied below. (i) Efficiecy of a series system: 98 ISSN 1927-0607 E-ISSN 1927-0615
Figure 4. Arragemet of Efficiecies i a Series system Figure 4 shows compoets with efficiecies E 1, E 2 etc. arraged i series. It ca easily be derived that the overall efficiecy E or the system is equal to the product of the compoet efficiecies, i.e., E= E 1.E 2 E = (2) i = 1 (ii) Efficiecy of a parallel system: Figure 5 shows -compoets with efficiecies E 1, E 2 etc. arraged parallely. Figure 5. Arragemet of Efficiecies i a Parallel system The compoets i parallel will share the iput power ad let the power i differet compoets be P 1, P 2 etc. The, the output from each compoets will be E 1 P 1, E 2 P 2 etc. so that P 0 the output power ca be expressed as follows, Where P EP. 0 i i i 1 i i P0 i 1 E, the system Efficiecy = (3) Pi Pi Whe greater part of the iput power flows through the compoets of higher efficiecy resultig efficiecy E of the system will be high ad vice-versa. O the other had if the iput power is shared equally amog all the compoets, E P 1 E E i, (4) For the efficiecy values of gear pairs which are high i practice, the system efficiecy is ot very much sesitive to the power sharig amog differet circuits or gear pairs. Hece the system efficiecy ca be take approximately equal to the average of the compoet efficiecies, eq. (4). (iii) Mixed System: I a mixed system, the combied efficiecy of the compoets i series ca be pairs such as (2-3) ad (2-4) i figure i 1 Published by Caadia Ceter of Sciece ad Educatio 99
2 should be take i series so that their combied efficiecy is (E 23, E 24 ). Gear pairs (1-3) ad (2-4) i Figure.3 are to be cosidered i parallel. Notig the remarks made i the estimate of efficiecy of a parallel system their combied efficiecy will be 1/2 (E 13 + E 24 ). Now, let us cosider some gear trais with five elemets (vertices) i.e. gear trais with three gear pairs. Simplest amog them is show i figure 6. The compoet efficiecy of the gear trai will be as give below. Figure 6. Gear trais with 5 elemets First Type E =E 23.E 24. E 35 (5) For compariso, aother gear trai with a mixed arragemet i.e., both parallel ad series, of gear wheels is show i figure 7 the schematic diagram is show i figure 7a while the correspodig graph is show i figure 7b. Figure 7 gear trais with five elemets Secod Type As suggested earlier, the gears 2, 3 ad 4 are cosidered i series so that their E 23, E 24. Now, the combied system of gear pairs (4-2-3) should be ½( E 15 + E 23.E 24 )j (6) For a clear uderstadig let us cosider gear trais with 6-elemets or four gear pairs Figure 8 shows the graphs of such gear trais. With the uderstadig we have from the previous examples oe ca see, figure 8(b), that the gear pairs (2-3) ad (2-4) are i series ad so is the case with gear pairs (1-5) ad (5-6). The assembly of the wheels (4-2-3) is the cosidered to be parallel with the assembly of the wheels (1-5-6). Now the efficiecy of the gear trai ca be writte directly. 100 ISSN 1927-0607 E-ISSN 1927-0615
Figure 8 Gear trais with 6 elemets-first Type E= ½ [(E 23.E 24 ) + (E 15.E 56 )] (7a) For the gear trai, figure 8a, the efficiecy E=1/2 [(E 15 + E 23.E 24.E 36 )] As a last example let us cosider aother gear trai which is slightly complicated, figure 9 (a & b) (7b) Figure 9. Gear trais with 6 elemets-secod Type Cosider the cluster of the gear pairs (2-3), (2-4) ad (2-6). The above arragemet suggests that of the three gear pairs, if pairs (2-3) ad (2-4) are cosidered i series the the pair (2-6) is parallel to the. Similarly, the pair (2-3) ad (2-6) ca be cosidered i series while the third pair (2-4) is i parallel with them. I geeral, ay two pairs ca be take i series while the third pair is i parallel with them. The efficiecies of all the possible combiatios are give below. Case (i) Pairs (2-3) ad (2-4) are i series ad pair (2-6) is i parallel. The Ec = The efficiecy of the cluster of gear wheels 2, 3, 4 ad 6 =1/2 (E 23.E 24 +E 26 ) (8) Case (ii) Gear Pairs (2-3) ad (2-6) are i series ad pair (2-4) is i parallel. The Ec = ½(E 23.E 26 +E 26 ), (9) Case (iii): Pairs (2-4) ad (2-6) are i series ad the pair (2-3) is i parallel. The Ec=1/2(E 24.E 26 +E 23 )j (10) I such a case it is safer to take the lowest value of the above three efficiecies, eq.(8-10) ad let this be E mi. Now, the et efficiecy of the gear trai =E=1/2(E mi +E 15 ) Published by Caadia Ceter of Sciece ad Educatio 101
6. Coclusios Assigmet of equal efficiecies to all the gear pairs eables the compariso of distict structures. For example structure figure. 3 is better tha figure. 2. Similarly, the gear trai, figure.7 is superior to figure.6. Specifyig the umber of teeth o the gear wheels eables estimatio of the aticipated efficiecy of the gear trai. I gear trais with equal umber of gear pairs, the gear trais with parallely arraged pairs are more efficiet. Parallely arraged serial pairs, figure.8(b), appear to be iferior to the arragemet i which more pairs arraged i series which i tur are parallel with the remaiig gear pairs (ot i series), figure.8(a). Gear trais with a cluster of wheels figure.9 is better tha the trais figure.8. Refereces C. H. Hsu, & J. J. Hsu. (1997). A efficiet methodology for the structural sythesis of geared kiematic chais. Mech. Mach. Theory, 1997, vol. 32, pp. 957-973. http://dx.doi.org/10.1016/s0094-114x(96)00081-x F. Buchsbaum, & F. Freudestei. (1970). Sythesis of kiematic structure of geared kiematic chais ad other mechaisms. J. Mechaisms,vol. 5, pp. (357-392). http://dx.doi.org/doi:10.1016/0022-2569(70)90068-6 J. E. Shigley, Miscke. (2003). Mechaical Egieerig Desig (SI Uits), Sixth editio, Tata Mc Graw Hill, p. 870. L. W. Tsai. (1989). O the Coceptual Desig of a Novel Class of Robot Cofiguratios, Tras ASME J. Mech. Tras. Auto. Desig, vol. 109, pp. 329-336. http://dx.doi.org/10.1115/1.3258798 M. F. Spotts. Mechaical Desig Aalysis, Pretice-Hall, Eagle wood cliffs, NJ, pp. 187-188. R. Ravisaker, & T. S. Mruthyujaya. (1985). Computerized sythesis of the structure of geared kiematic chais. Mech. Mach. Theory, vol. 20, pp. 367-387. http://dx.doi.org/10.1016/0094-114x(85)90042-4 102 ISSN 1927-0607 E-ISSN 1927-0615