weight design

Mechanism and Machine Theory 73 (2014) 197–217

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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt

Gear train optimization based on minimum volume/weight design Sa'id Golabi a, Javad Jafari Fesharaki b,⁎, Maryam Yazdipoor b a b

Department of Mechanical Engineering, University of Kashan, Kashan, Iran Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran

a r t i c l e

i n f o

Article history: Received 21 April 2013 Received in revised form 4 November 2013 Accepted 8 November 2013 Available online 8 December 2013 Keywords: Gearbox design Optimization Minimum volume/weight Gear train

a b s t r a c t In this study, the general form of objective function and design constraints for the volume/ weight of a gearbox has been written. The objective function and constraints can be used for any number of stages for gearbox ratio but in this paper one, two and three-stage gear trains have been considered and by using a Matlab program, the volume/weight of the gearbox is minimized. Finally, by choosing different values for the input power, gear ratio and hardness of gears the practical graphs from the results of the optimization are presented. From the graphs, all the necessary parameters of the gearbox such as number of stages, modules, face width of gears, and shaft diameter can be derived. The results are compared with those reported in the previous works and an example is presented to show how the practical graphs can be used. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction Gear trains are the most common of machine components and the problem of their minimum weight or minimum volume design has been a subject of many researches. By integrating the configuration design process and dimensional, Chong et al. [1] suggested a new generalized method and algorithm to design multi stage drives. Their suggested algorithm consists of four stages. At the first, the user considered the number of reduction stages provisionally. Next, by using the random search method the gear ratio of each stage is specified. Third by using generate and test methods, the basic gear parameters are chosen. At the end, by using the simulated annealing algorithm, the values of other design parameters are defined. The objective function that they considered is minimizing the geometrical volume. Prayoonrat and Walton [2] depicted an algorithm to optimize and design multi spindle gear trains. In their algorithm the designer could choose many options to optimize the gear trains such as minimum volume or minimum overall size. Wang et al. worked on the optimal engineering design of spur gear sets and tooth profile [3,4]. By using genetic algorithm, Yokota et al. formulated an optimization problem for the weight design of a gear [5]. They considered the bending strength of gear and shafts gear dimensions as constrains of the optimization problem. Using interactive physical programming Huang et al. investigate the multi objective optimization reduction units with three-stage spur gear [6]. Pomrehn and Papalambros worked on discrete optimal design formulations with application to gear train design [7]. Gologlu and Zeyveli depicted a genetic approach to automate a preliminary design of gear drives [8]. Thompson et al. [9] worked on minimizing the volume of single and multi stage spur gear reduction units. Based on design criteria their method is applied to the units with two-stage and three-stage spur gear

⁎ Corresponding author at: Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Isfahan, Iran. Tel.: +98 9133167153. E-mail address: [email protected] (J.J. Fesharaki). 0094-114X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmachtheory.2013.11.002

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Nomenclature b Di d dw1 el eh ew Ft h KB KH KO KS KV Lin Lout l n Ma mg mp mt Ng Np R r Se Sf s Sfs SH Sy Tm t ui w YJ YN YZ YV ZI ZE ZN ZR ZW β φt σFP σHP

Net face width Diameter of gear Shaft diameter Operating pitch diameter of pinion (mm) Gap for gear and shell in length direction Gap for gear and shell in height direction Gap for gear and shell in width direction Transmitted tangential load (N) Height of gearbox Rim thickness factor Load distribution factor Over load factor Size factor Dynamic factor Length of input shaft Length of output shaft Length of gearbox Number of shafts Alternating bending moment-shaft design Metric module of gear Metric module of pinion Transverse metric module (mm) Number of teeth on gear Number of teeth on pinion Overall reduction ratio of gearbox Radius of gear Endurance limit-shaft design Safety factor-bending Number of stages Safety factor-shaft design Safety factor-pitting Yield strength-shaft design Midrange torque of shaft Thickness of shell of gearbox Partial reduction ratio of each gearbox stages (i = 1, 2, 3) Width of gearbox Geometry factor for bending strength Stress cycle life factor for bending strength Reliability factor Temperature factor Geometry factor for pitting resistance Elastic coefficient (N/mm2)0.5 Stress cycle life factor for pitting resistance Surface condition factor for pitting resistance Hardness ratio factor for pitting resistance Helix angle at standard pitch diameter Transverse pressure angle Allowable bending stress number (N/mm2) Allowable contact stress number (N/mm2)

reduction. Petre et al. worked on the design and simulation of a steering gearbox with variable transmission ratio [10]. By using the genetic algorithm optimization of the modulus of spur gears, the diameter of shafts and rolling bearing is investigated by Mendi et al. [11]. Their procedure is based on minimizing the volume of the gearbox. Pi worked on minimizing the gearbox length of a four step helical gearbox [12]. By using Steady State Finite Element Analysis Joule et al. [13,14] investigated the Thermal

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Fig. 1. Configuration of gearbox and parameters.

Analysis of a Spur Gearbox. By using the genetic algorithm Hyong and Song [15] introduced a design method to optimize gear trains. They investigated the minimizing of the volume of two stage gear trains to show that for solving the discrete, integer variable and continuous problem, the genetic algorithm is better than other conventional algorithms. Based on the random search method, Zarefar and Muthukrishnan [16] investigated the optimization of helical gear design. Savsani et al. [17] used particle swarm and simulated annealing algorithms to optimizing the weight of a multi stage gear train. For the reduction of gear size and meshing vibration Chong et al. [18] worked on multi objective optimal design of cylindrical gear pairs. With numerical methods Ciavarella and Demelio [19] investigated the optimization of stress concentration, specific sliding and fatigue life of gears. Swantner and Campbell described a method for automating the design of gear trains comprised of simple, compound, bevel and worm gears [20]. Deb and Jain worked on a multi speed gearbox design using multi objective evolutionary algorithms [21]. Based on a selection of optimal materials, optimal position of shaft axes and optimal gear ratio, Marjanovic et al. [22] provided a description for selection of the optimal gearbox.

Fig. 2. Two possible configurations of 3-stage gearbox [22].

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Table 1 Comparison between results of presented paper and those reported by Chong and Lee [26]. Reported by Chong and Lee [26]

Presented paper

Description

F: Volume of material of gears

F: Volume of material of gears

m1 m3 N1 N3 N2 N4 F1 F2 Volume of gears (mm3) Difference

1 2.5 18 14 134 54 30 40.1 1377539 −7.79%

1.5 3 16 14 109 58 15.30 30.60 1270200

In this paper the overall volume of gear train is minimized. For this purpose the general form of objective function and design constraints has been written. These equations can be used for any number of stages in a gearbox but in this paper one, two and three-stage gear trains have been considered and by using a Matlab program for objective function and constraints, the practical graphs from the results is presented. Next, the results are compared and validated with those reported in the previous publications. At last an example is considered to show how the results of this paper and graphs can be used. 2. Problem statement — Objective function To find the optimum gearbox based on volume/weight, the objective function and constrains must be specified at first. But because all components in a gear box have a density near the steel, so the volume function for the gearbox components can be considered as the weight function of gearbox according to the density relative. So the objective function is formulated as the volume of materials of gearbox and can be written as the sum of the volume of gears, shafts and shell. Volume of materials ¼ mShafts þ mGears þ mShell

ð1Þ

And the components of this equation are shown in Eqs. (2) to (4):

mShafts ¼

mGears ¼

n −1 X π  d21 π  d2n π  d2i  ðw þ Lin Þ þ  ðw þ Lout Þ þ w 4 4 4 i¼2 2S X π  D2 i

i¼1

4

 bi −

n −1 X i¼1

ð2Þ

π  d2i π  d21 π  d2n  ðb2i−2 þ b2i−1 Þ‐  b1 ‐  b2S 4 4 4

ð3Þ

Table 2 Comparison between results of presented paper and those reported by Marjanovic et al. [22]. 3 Stages

2 Stages

Description

Ref. [22]

Presented paper

Ref. [22]

Presented paper

m1 m2 m3 N1 N3 N5 N2 N4 N6 F1 F2 F3 Length of gears Difference

2.49 3.25 3.70 20.09 22.33 27.51 71.79 85.92 76.93 76.46 109.48 146.19 681.24 −7.24%

3.31 5.05 7.89 13 13 13.00 54.50 50.18 32.13 51.97 79.38 123.97 631.89 −4.85%

2.07 4.67 – 22.69 18.32 – 185.48 86.04 – 65.00 107.06 – 714.00

3.19 6.30 – 13 13 – 117.18 57.69 – 50.12 98.92 – 679.38

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Table 3 Selected values of input data. Input parameter

Selected values

Transmission power Hardness of material Gearbox ratio

2, 5, 10, 20, 30, 50, 80, 100, 150, 200 hp 200, 300, 400 BHN 1.5, 2, 3, 5, 8, 10, 15, 20, 40, 50

mShell ¼ ðw  l  hÞ−½ðw−2t Þ  ðl−2t Þ  ðh−2t Þ

ð4Þ

where, l, h and w are the length, height and width of gearbox, respectively and for a gearbox, these parameters can be presented as: l¼

2S D1 D2S X Di þ þ þ ð2el Þ þ ð2t Þ 2 2 2 i¼1

h ¼ D max þ ð2eh Þ þ ð2t Þ w ¼ x max þ

b þ ew þ t 2

ð5Þ ð6Þ ð7Þ

where “x” is the horizontal position of gears and shows the horizontal distance of the center of gear from shell and “s” is the number of shafts in gearbox. All other parameters are explained in Fig. 1.

Fig. 3. Flowchart of determination of design parameters by using the obtained curves.

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Fig. 4. Optimal number of stages of gearbox, (a) comparison of 1 and 2 stage gearbox; (b) comparison of 2 and 3 stage gearbox; (c) comparison of 3 and 4 stage gearbox.

According to consideration of the previous parameters, the position of the first pair of gears, only depends on the thickness of shell and face width of gears, and the second pair is placed at a horizontal distance from the first pair. For the third and next pairs, there are various conditions for locating the gears. If it is possible, gear pairs are placed between the previous gears, and optimal configuration may be selected from possible configurations. For example in Fig. 2, the 3-stage gearbox is presented. If gear number 5 has no interference with second gear, the third pair can be placed on the left of the previous pair (Fig. 2-A), but if gear number 5 has any interface with previous gears such as gear number 2, the gear is located after all previous gears (Fig. 2-B).

3. Problem statement — Constraints To optimize the gearbox volume/weight, three kinds of constraints have been considered.

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Fig. 5. Optimal partial ratio of 2-stage gearbox.

3.1. Geometrical constraint A geometrical constraint has been considered to avoid the interference between each gear and the next shaft and in general form for all gears it can be written as: r 2i br 2iþ1 þ r 2iþ2

ð8Þ

where, “r” is the radius of gear. For example for the first stage of gearbox Eq. (8) can be written as: r 2 br3 þ r 4 :

ð9Þ

The second geometrical constraint has been considered to locate the best position for gears and it can be written as: r 2i þ r 2iþ3 br 2iþ1 þ r 2iþ2 :

ð10Þ

3.2. Design constraints Design constraints for gearbox component are considered as bending strengh of gears, pitting resistance of gears and strength of shafts based on maximum shear stress theory. The equations of these constraints are presented in Eqs. (11), (12) [23] and 13 respectively. σ Bending bσ Bending−allowable →F t  K O  K v  K S 

K H  K B σ FP Y N b  b  mt  Y J S f Y θ Y Z

Fig. 6. Optimal partial ratio of 3-stage gearbox.

ð11Þ

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Fig. 7. Design parameters of 1-stage gearbox for H = 200 BHN.

 σ Contact bσ Contact−allowable →Z E F t  K O  K v  K S 

KH Z  R dw1  b Z I

1=2

b

σ HP  Z N  Z W SH  Y θ  Y Z

8 " !2   #1=2 91=3 <32S = Tm Ma 2 FS þ : d≥ : π ; Sy Se

ð12Þ

ð13Þ

3.3. Control parameter constraints The third kind of constraints are considered to control the value of parameters and are listed as: Minimum tooth number of each pinion [24]:

Np ≥

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi1 0 !2 !! u Ng u Ng Ng 2 cosðβ2i−1 Þ t 2 @ ! þ þ 1þ2 sin ðφt ÞA: Np Np Np Ng 2 1þ2 sin ðϕt Þ Np

ð14Þ

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Fig. 8. Design parameters of 1-stage gearbox for H = 300 BHN.

Maximum gear tooth number: Ng ≤120:

ð15Þ

Gear face width: 3  π  mt bFb5  π  mt :

ð16Þ

For reduction the ratio of gearbox: Np bNg :

ð17Þ

Equality of modulus of each pair: mp bmg :

ð18Þ

Modulus constant number: 1≤mp ≤50:

ð19Þ

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Fig. 9. Design parameters of 1-stage gearbox for H = 400 BHN.

All the above relations should be written in general form to establish for all stages in gearbox and the speed ratio of gearbox “R” is the multiplying of the speed ratio of each stage of gearbox and in general form can be expressed as: S−1 N

∏

i¼1

2i−1

N2i

¼ R:

ð20Þ

4. Optimization method The optimization of gearbox cannot be carried out without computer programming. In this paper, the optimization is performed using fmincon (Find minimum of constrained nonlinear multivariable function) which is a component of the MATLAB optimization toolbox. This algorithm, which implements constraints optimization on nonlinear multivariable problems exhibits generally good performance on a wide range of problems [25]. For this purpose, 2 M-files containing objective function and constraint functions are written that include over 500 command lines. 5. Results validation To confirm the results, two problems that published previously are considered. In the first problem, the parameters of gearbox in the program are changed to those considered by Chong and Lee [26]. They used Genetic Algorithm in minimizing a two-stage gear train. Table 1 shows the comparison between the results of two procedures. For complete similarity the results of the

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Fig. 10. Design parameters of first stage of 2-stage gearbox for H = 200 BHN.

objective function in our program is changed to optimizing the gears mass. It is observed that the results of the present paper has a reduction of about 7.8% compared to the results reported by Chong and Lee. In the second example the input data and bound variables of problem are changed to those considered by Marjanovic et al. [22]. They considered the length of the gear train as the objective function. The objective function is the length of location gears in gearbox, and so some parameters such as thickness of shell of gearbox have zero values in program. Table 2 shows the comparison between the results from the presented paper and that reported by Marjanovic. It is seen that the results show about 4.8% and 7.2% reduction for a gearbox with two or three stage ratio reduction respectively. 6. Results and useful charts By considering Eq. (1) as the objective function and using Eqs. (2) to (7), and considering Eqs. (8) to (19) as constraints, the optimization program can present the value of objective function and all other values for gearbox parameters such as module etc. In changing the results of the computer program to useful charts, more than 600 problems are solved for several values of transmission power, hardness of material and gearbox ratio. The values that considered for power, hardness and gearbox ratio, are presented in Table 3. After solving the problems, the results are changed into applicable curves that specify all the necessary parameters of gearbox design. In using the curves consider the presented flowchart in Fig. 3. First, with the overall ratio of gearbox, the number of stages of gearbox is obtained from Fig. 4. In this figure, mass of materials of gearbox is plotted versus gearbox ratio. So for a given ratio, number of stages which has the lowest mass should be selected. After selecting the number of stages, the partial ratios of gearbox can be obtained from Figs. 5 or 6 according to those obtained from Fig. 4. Then, as the flowchart shows, depending on hardness of

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Fig. 11. Design parameters of first stage of 2-stage gearbox for H = 300 BHN.

material, design parameters of each stage of gearbox can be achieved from related curves in Figs. 7 to 24. Figs. 7, 8 and 9 display the values of design parameters for 1 stage gearbox with 200, 300 and 400 BHN respectively. Figs. 10 to 15 show the values of design parameters for 2 stage gearbox with 200, 300 and 400 BHN respectively. In these figures, Figs. 10, 11 and 12 specify the values for gears mounted on the first and second shafts and Figs. 13, 14 and 15 show the values for gears mounted on the second and third shafts. Figs. 16 to 24 show the values of design parameters for 3 stage gearbox with 200, 300 and 400 BHN respectively. Figs. 16, 17 and 18 specify the values for gears in the first stage of gearbox. Figs. 19, 20 and 21 show the design parameters for the second stage of gearbox and Figs. 22, 23 and 24 show the design parameters for third stage of gearbox. In each figure, by using transmitted power and partial ratio of the stage, the module, face with of gear pair and diameter of shaft can be determined.

7. Example This section presents an example of selecting of design parameters of gearbox with optimal weight using curves obtained in the present paper. Consider the input data as: Total ratio = 15, Power = 15 hp, hardness of material = 400 BHN. As previously mentioned, the optimal number of stages of gearbox is obtained from Fig. 4-b. This figure shows that 3-stage gearbox is the optimal selection for the given transmission ratio.

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209

Fig. 12. Design parameters of first stage of 2-stage gearbox for H = 400 BHN.

Also from Fig. 6, optimal partial ratios of 3-stage gearbox for the total ratio 15, are: u1 ¼ 2:6;

u2 ¼ 2:6;

u3 ¼ 2:2:

Since the hardness of material is 400 BHN, the design parameters of the first, second and third stages should be selected from Figs. 18, 21 and 24 respectively. In these figures partial ratios that were obtained from the previous step specify that curves with hollow square should be used. By using value of transmission power, the design parameter of each stage can be determined from related curve. These values are listed in Table 4 as: 8. Conclusion In this study the general form of objective function and design constraints for the volume of a gear train has been written and by using the Matlab program, the overall volume of one, two and three-stage gear trains is minimized. Next by considering some values for transmission power, hardness of material and gearbox ratio as input data for gearbox parameters, the results from the optimization program have been presented in the form of practical curves. The practical curves can be used to specify all the necessary parameters of gearbox such as number of stages, gear modules, face width of gears and shaft diameters. The results of this paper are compared and validated with those reported in the previous publication. Then, an example is presented to show how the curves can be used.

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Fig. 13. Design parameters of second stage of 2-stage gearbox for H = 200 BHN.

Fig. 14. Design parameters of second stage of 2-stage gearbox for H = 300 BHN.

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Fig. 15. Design parameters of second stage of 2-stage gearbox for H = 400 BHN.

Fig. 16. Design parameters of first stage of 3-stage gearbox for H = 200BHN.

211

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Fig. 17. Design parameters of first stage of 3-stage gearbox for H = 300BHN.

References [1] T.H. Chong, I. Bae, G.-J. Park, A new and generalized methodology to design multi-stage gear drives by integrating the dimensional and the configuration design process, Mech. Mach. Theory 37 (2002) 295–310. [2] S. Prayoonrat, D. Walton, Practical approach to optimum gear train design, Comput. Aided Des. 20 (1988) 83–92. [3] H. Wang, H.-P. Wang, Optimal engineering design of spur gear sets, Mech. Mach. Theory 29 (1994) 1071–1080. [4] Y. Wang, Optimized tooth profile based on identified gear dynamic model, Mech. Mach. Theory 42 (2007) 1058–1068. [5] T. Yokota, T. Taguchi, M. Gen, A solution method for optimal weight design problem of the gear using genetic algorithms, Comput. Ind. Eng. 35 (1998) 523–526. [6] H.-Z. Huang, Z.-G. Tian, M. Zuo, Multiobjective optimization of three-stage spur gear reduction units using interactive physical programming, J. Mech. Sci. Technol. 19 (2005) 1080–1086. [7] L.P. Pomrehn, P.Y. Papalambros, Discrete optimal design formulations with application to gear train design, J. Mech. Des. 117 (1995) 419–424. [8] C. Gologlu, M. Zeyveli, A genetic approach to automate preliminary design of gear drives, Comput. Ind. Eng. 57 (2009) 1043–1051. [9] D.F. Thompson, S. Gupta, A. Shukla, Tradeoff analysis in minimum volume design of multi-stage spur gear reduction units, Mech. Mach. Theory 35 (2000) 609–627. [10] P. Alexandru, D. Macaveiu, C. Alexandru, Design and simulation of a steering gearbox with variable transmission ratio, Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 226 (2012) 2538–2548. [11] F. Mendi, T. Baskal, K. Boran, F.E. Boran, Optimization of module, shaft diameter and rolling bearing for spur gear through genetic algorithm, Expert Syst. Appl. 37 (2010) 8058–8064. [12] V.N. Pi, Optimal calculation of partial transmission ratios for four-step helical gearboxes with first and third step double gear-sets for minimal gearbox length, Proceedings of the American Conference on Applied Mathematics, World Scientific and Engineering Academy and Society (WSEAS), Cambridge, Massachusetts, 2008, pp. 29–32. [13] D. Joule, S. Hinduja, J.N. Ashton, Thermal analysis of a spur gearbox part 1: steady state finite element analysis, Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 202 (1988) 245–256. [14] D. Joule, S. Hinduja, J.N. Ashton, Thermal analysis of a spur gearbox part 2: transient state finite element analysis of the gearbox casing, Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 202 (1988) 257–262. [15] C. Tae Hyong, L. Joung Sang, A design method of gear trains using a genetic algorithm, Int. J. Precis. Eng. Manuf. 1 (2000) 62–70.

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Fig. 18. Design parameters of first stage of 3-stage gearbox for H = 400 BHN.

[16] H. Zarefar, S.N. Muthukrishnan, Computer-aided optimal design via modified adaptive random-search algorithm, Comput. Aided Des. 25 (1993) 240–248. [17] V. Savsani, R.V. Rao, D.P. Vakharia, Optimal weight design of a gear train using particle swarm optimization and simulated annealing algorithms, Mech. Mach. Theory 45 (2010) 531–541. [18] T.H. Chong, I. Bae, A. Kubo, Multiobjective optimal design of cylindrical gear pairs for the reduction of gear size and meshing vibration, JSME Int. J. Ser. C Mech. Syst. Mach. Elem. Manuf. 44 (2001) 291–298. [19] M. Ciavarella, G. Demelio, Numerical methods for the optimisation of specific sliding, stress concentration and fatigue life of gears, Int. J. Fatigue 21 (1999) 465–474. [20] A. Swantner, M.I. Campbell, Topological and parametric optimization of gear trains, Eng. Optim. (2012) 1–18. [21] K. Deb, S. Jain, Multi-speed gearbox design using multi-objective evolutionary algorithms, J. Mech. Des. 125 (2003) 609–619. [22] N. Marjanovic, B. Isailovic, V. Marjanovic, Z. Milojevic, M. Blagojevic, M. Bojic, A practical approach to the optimization of gear trains with spur gears, Mech. Mach. Theory 53 (2012) 1–16. [23] A.A.S. 2101-C95, Fundamental rating factors and calculation methods for involute spur and helical gear teeth, American Gear Manufacturers Association1995. [24] R.G. Budynus, J.K. Nisbett, Shigly's Mechanical Engineering Design, 9th ed., 2011. [25] M.A. Branch, A. Grace, MATLAB Optimization Toolbox User's Guide, The Math Works, Natick, MA, 1996. [26] T.H. Chong, J.S. Lee, A design method of gear trains using a genetic algorithm, Int. J. Korean Soc. Precis. Eng. 1 (2000).

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Fig. 19. Design parameters of second stage of 3-stage gearbox for H = 200 BHN.

Fig. 20. Design parameters of second stage of 3-stage gearbox for H = 300 BHN.

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Fig. 21. Design parameters of second stage of 3-stage gearbox for H = 400 BHN.

Fig. 22. Design parameters of third stage of 3-stage gearbox for H = 200 BHN.

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Fig. 23. Design parameters of third stage of 3-stage gearbox for H = 300 BHN.

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Fig. 24. Design parameters of third stage of 3-stage gearbox for H = 400 BHN.

Table 4 Example results from presented charts. Total ratio = 15, Power = 150 hp, hardness of material = 400 BHN u1 = 2.6, u2 = 2.6, u3 = 2.2

First stage Second stage Third stage

No. of teeth

Module

Face width (mm)

Shaft dia. (mm)

17 12 12

5 8 10

51 82 170

35 43 125, 72

Volume of material = 2.6 × 10 7 mm 3.