A STUDY ON CALCULATING OPTIMUM GEAR RATIOS OF A THREE-STAGE HELICAL GEARBOX

Transcription

1 International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue 11, November 018, pp , Article ID: IJMET_09_11_14 Available online at ISSN Print: and ISSN Online: IAEME Publication Scopus Indexed A STUDY ON CALCULATING OPTIMUM GEAR RATIOS OF A TREE-STAGE ELICAL GEARBOX Vu Ngoc Pi Thai Nguyen University of Technology, Thai Nguyen city, Vietnam Tran Thi ong Nguyen Tat Thanh University, o Chi Minh city, Vietnam Le ong Ky Vinh Long University of Technology Education, Vietnam Le Xuan ung Thai Nguyen University of Technology, Thai Nguyen city, Vietnam Luu Anh Tung Thai Nguyen University of Technology, Thai Nguyen city, Vietnam Nguyen Khac Tuan Thai Nguyen University of Technology, Thai Nguyen city, Vietnam ABSTRACT This article presents a study on the calculation of optimum gear ratios of a threestage helical gearbox. In the study, for determining the optimum gear ratios, an optimization problem was performed. Besides, the gearbox cross section area was chosen as the objective function of the problem. Moreover, a simulation experiment was designed and performed by a computer program for evaluating the effect of input parameters on the optimum gear ratios. As a result, the influence of the input factors on the optimum gear ratios were learned and models for determination of the optimum gear ratios were proposed. Key words: Gear ratio, optimum gearbox design, three-stage helical gearbox. Cite this Article: Vu Ngoc Pi, Tran Thi ong, Le ong Ky, Le Xuan ung, Luu Anh Tung and Nguyen Khac Tuan, a Study on Calculating Optimum Gear Ratios of a Three-Stage elical Gearbox, International Journal of Mechanical Engineering and Technology, 9(11), 018, pp editor@iaeme.com

2 Vu Ngoc Pi, Tran Thi ong, Le ong Ky, Le Xuan ung, Luu Anh Tung and Nguyen Khac Tuan 1. INTRODUCTION In gearbox design, the determination of the optimum gear ratios has a very important role. This is because the gear ratios powerfully affect the size, the mass, and therefore the cost of the gearbox. Consequently, the calculation of the optimum gear ratios has been subjected to many studies. Until now, the gear ratios have been determined for helical gearboxes with two stages [1,, and 4], three stages [4, 5, 6, 7 and 8], and four stages [4, 8, 9 and 10]. Besides, the gear ratios can be determined by using several methods including the graph method [1, and 4], the practical method [] and the model method [5, 6, 7, 8, 9 and 10]. Using the graph method, the gear ratios is determined graphically. For instance, the gear ratios of the first stage and the second stages of a three-stage helical gearbox can be found from Figure 1 [1]. Using the practical method, the gear ratios can be determined by practical data. For example, based on the data from gearbox factories, it was noted that the weight of a three-stage gearbox is minimum when the center distance ratio of the second stage to that of the first stage was from 1.4 to 1.6 []. The optimum gear ratios were therefore introduced in the tabulated form. Using the model method, equations for calculating the gear ratios have been proposed for different objectives. The objectives can be the minimum volume of gears [4], the minimum gearbox cross section area [5], the minimum gearbox length [7] or the minimum gearbox mass [8]. Figure 1. Partial gear ratios versus the total gear ratio [1] This paper presents a study on the optimum calculating the gear ratios of a three-stage helical gearbox with the objective is the minimum gearbox cross section area. Also, the influence of the input factors on the optimum gear ratios was evaluated.. OPTIMIZATION PROBLEM Figure. Calculation schema 19 editor@iaeme.com

3 A Study on Calculating Optimum Gear Ratios of a Three-Stage elical Gearbox The gearbox cross section area is determined as (see Figure ): A = L h (1) In which, L and h are calculated as: L = d / + a + a + d / w11 w w w h = max d, d, d w1 w w Also, d w and dw can be determined as [6]: w11 w1 1 d = a / u + 1 w w d = a u / u + 1 d w = aw u / u + 1 (6) In the above equations, a w, aw are the center distance and d w1, d w, dw are the pitch diameters of the first, the second and the third stage; u are the gear ratios the second and the third stage, respectively. Therefore, the optimization problem can be defined by the following equation: Minimize A = L h (7) With two following constraints () () (4) (5) 1 u 9 (8) 1 u 9 From (), (), (5), (6) and (7), it is clear that for solving the optimization problem it is necessary to determine d w11, d w1, d w, d w, a w1, a w and a w..1. Calculating the center distance and the pitch diameters of the first stage The center distance of the first stage aw 1 is calculated by [6]: ( 1) a = k u + 11 w1 m 1 T k ψ 1 ba1 In which, k m is material coefficient; with the gear material is steel, m contact load ratio for pitting resistance (MPa); In practice, [ ] k = [11]; [ ] (9) k =4 [11]; K is the σ is the allowable contact stress σ = K (MPa); ψ ba1 is coefficient of wheel face width of the first stage ψ = K [11]; of the gearbox; ba From the moment equilibrium condition of the mechanic system which includes three helical gear sets we have: T = T η η 4 out 11 hg b g (10) editor@iaeme.com

4 Vu Ngoc Pi, Tran Thi ong, Le ong Ky, Le Xuan ung, Luu Anh Tung and Nguyen Khac Tuan Where, ηhg is helical gear transmission efficiency ( η hg = 0.96K 0.98 [11]); ηb is transmission efficiency of a pair of rolling bearing ( η b = 0.99K [11]). Choosingη hg = 0.97 and η b = 0.99 gives T11 = Tout / u g Choosing k = 1.1 and substituting (11) and into (9) with the note that u u = ug / u1 gets: a Tout = ( u + 1) ψ w1 1 g 1 ba1 Therefore, the pitch diameters of the first stage are calculated as [11]: w11 w1 1 d = a / u + 1 w 1 w1 1 1 d = a u / u + 1 (11) (1) (1) (14).. Calculating the center distance and the pitch diameter of the second stage For the second stage, the center distance aw is determined as [11]: T1 k aw = km ( u + 1) [ σ ] ψ ba (15) Wherein, K is contact load ratio for pitting resistance; k = [11] and we can chose k = 1.1; ψ ba is coefficient of wheel face width of the second stage of the gearbox; ψ ba = 0.K 0.5 [11]; In addition, we have: T = T η η out 1 hg b Choosing 0.97 hg η = and 0.99 T = T / u 1 out η = (as in Section.1) we have b Substituting (17) and k = 1.1 into (15) with the note that u u = ug / u1 gets: a Tout 1 = ( u + 1) ψ w g ba The pitch diameter of the second stage then is determined by [11]: w w d = a u / u + 1 (16) (17) (18) (19).. Calculating the center distance and the pitch diameter of the third stage The center distance of the third stage aw can be determined by the following equation [6]: ( 1) a = K u + 1 w m T k ψ ba (0) editor@iaeme.com

5 A Study on Calculating Optimum Gear Ratios of a Three-Stage elical Gearbox Also, for the third stage we have T = T η η out 1 hg b Choosing 0.97 hg T = T / u a 1 out η = and 0.99 η = as in section.1 gives b Substituting (), k m =4 and k = 1.1 (as in section.1) into (0) we have: Tout = ( u ) w u ψ ba The pitch diameter of the third stage then is determined by [11]: w w d = a u / u + 1 (1) () () (4).4. Experimental work For evaluating the effect of the input factors on the optimum gear ratios, a simulation experiment was designed and performed by a computer program. Also, a -level full factorial design was selected for this experiment. In addition, 6 input factors were chosen for the exploring (Table 1). 6 Therefore, the design was arranged with = 64 number of experiments. To perform the experiment, a computer program was constructed based on equations (7) and (8). Table shows the various levels of input factors and the output responses (the optimum gear ratios of the second and the third stage u ). Table 1. Input factors Factor Code Unit Low igh Total gearbox ratio u g Coefficient of wheel face width of stage 1 x ba Coefficient of wheel face width of stage x ba Coefficient of wheel face width of stage x ba Allowable contact stress AS MPa Output torque T out Nmm Table. Experimental plans and output responses StdOrder RunOrder CenterPt Blocks u g K be Xba1 Xba AS (MPa) Tout (Nm) u 1 u editor@iaeme.com

6 Vu Ngoc Pi, Tran Thi ong, Le ong Ky, Le Xuan ung, Luu Anh Tung and Nguyen Khac Tuan. OPTIMIZATION RESULTS AND DISCUSSIONS Figure shows the main effect of each input factor on the optimum gear ratios of the second satge u (Figure.a) and the third stage u (Figure.b). From the figure, bothu increase considerably with the increase of the total gearbox ratio u g. Also, they effected by the wheel face width coefficients of three helical gear sets ( ψ ba1, ψ ba andψ ba ). Moreover, it was found that the optimum gear ratios are not affected by the allowable contact stress and the output torque. Figure 4 describes the Pareto chart of the standardized effects of the input factors. From this figure, the bars that represent factors including the total gearbox ratio (factor A), the wheel face width coefficients of three gear set (factors B, C and D) and the interactions between them cross the reference line. Subsequently, these factors are statistically significant at the 0.05 level with the response model. Figure 5 presents the Normal Plot of the standardized effects for u (Figure 5.a) (Figure 5.b). From the figure, the total gearbox ratio (factor A) is the most significant factor for both u. Also, factors A ( u g ) and D ( ψ ba ) has a positive standardized effect for both u while ψ ba1 and ψ ba (factors B and C) have a negative standardized effect. a) b) Figure. Main effects plot for u editor@iaeme.com

7 A Study on Calculating Optimum Gear Ratios of a Three-Stage elical Gearbox a) b) Figure 4. Pareto Chart of the Standardized Effects for u a) b) u Figure 5. Normal Plot for u and a) editor@iaeme.com

8 Vu Ngoc Pi, Tran Thi ong, Le ong Ky, Le Xuan ung, Luu Anh Tung and Nguyen Khac Tuan b) Figure 6. Estimated Effects and Coefficients for u Figure 6 shows the estimated effects and coefficients for u (Figure 6.a) (Figure 6.b). From the figure, factors which have a significant effect on a response have P-values lower than 0.05 are the total gearbox ratio u g, the wheel face width coefficients of three gear sets ( ψ ba1, ψ ba andψ ba ) and their interactions. Therefore, the following equations were created for the relation between the optimum gear ratios and the significant effect factors: u = ψ ψ ψ 0.05 ψ + g ba1 ba ba g ba ψ + 6 ψ ψ 6 ψ ψ g ba ba1 ba ba1 ba (4) u = ψ ψ ψ u ψ g ba1 ba ba g ba g ψ ba g ψ ba + 9 ψ ba1 ψ ba ψ ba ψ ba (5) The above proposed equations fit the data very well because the adj-r and pred-r are in the high values (Figure 6). Equations (4) and (5) are used to determine the optimum gear ratios of the second and the third stages u. After that, the optimum gear ratio of the first stage is easily calculated by u = u / u u. 1 g 4. CONCLUSIONS A study on determining the optimum gear ratios of a three stage helical gearbox for getting the minimum gearbox cross section area was carried out. In this study, the effect of the input factors including the total gearbox ratio, the wheel face width coefficients of three helical gear sets, the allowable contact stress and the output torque was inspected. Also, equations for determining the optimum partial gear ratios for getting the minimum gearbox cross section area were proposed. The optimum gear ratios can be calculated simply by using these explicit models. ACKNOWLEDGEMENTS The work described in this paper was supported by Thai Nguyen University of Technology for a scientific project editor@iaeme.com

9 A Study on Calculating Optimum Gear Ratios of a Three-Stage elical Gearbox REFERENCES [1] Кудрявцев В.Н., Державец Ю.А., Глухарев Е.Г. Конструкции и расчет зубчатых редукторов, Справочное пособие, Издательство: Л. Машиностроение, с. [] G. Milou; G. Dobre; F. Visa;. Vitila, Optimal Design of Two Step Gear Units, regarding the Main Parameters, VDI Berichte No 10 (1996), p. 7. [] Vu Ngoc Pi, A method for optimal calculation of total transmission ratio of two step helical gearboxes, Proceedings of the National conference on Engineering Mechanics, a Noi (001), p. 1. [4] A.N. Petrovski, B.A. Sapiro, N.K. Saphonova, About optimal problem for multi-step gearboxes (in Russian),Vestnik Mashinostroenie, No. 10 (1987), p. 1. [5] Vu Ngoc Pi, Nguyen Khac Tuan, Optimum determination of partial transmission ratios of Three-step helical gearboxes for getting minimum cross section, Journal of Environmental Science and Engineering A 5 (016) [6] Vu Ngoc Pi, Nguyen Dang Binh, Vu Quy Dac, Phan Quang the, Optimal Calculation of Total Transmission Ratio of Three-Step elical Gearboxes for Minimum Mass of Gears. (In Vietnamese) Journal of Science and Technology of 6 Engineering Universities (006), p. 91. [7] Vu Ngoc Pi, A new study on optimal calculation of partial transmission ratio of three-step helical reducers, The rd IASME / WSEAS International Conference on Continuum Mechanics, Cambridge, UK (008), p.. [8] Romhild I., Linke., Gezielte Auslegung Von Zahnradgetrieben mit minimaler Masse auf der Basis neuer Berechnungsverfahren, Konstruktion 44, 199, pp [9] Vu Ngoc Pi, Optimal determination of partial transmission ratios for four-step helical gearboxes with firt and third step double gear-sets for minimal mass of gears, Applied Computing Conference (ACC 08), Istanbul, Turkey, May 7-0, 008. [10] Le Xuan ung, Vu Ngoc Pi, Nguyen Van Du, Optimal calculation of partial transmission ratios of four-step helical gearboxes with second and fourth-step double gear-sets for minimal mass of gears, The international symposium on Mechanical Engineering (ISME o Chi Minh city, Vietnam, 1-, September 009. [11] Trinh Chat, Le Van Uyen, Design and calculus of Mechanical Transmissions (in Vietnamese), Educational Republishing ouse, anoi, editor@iaeme.com