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A STUDY ON CALCULATION OF OPTIMUM GEAR RATIOS FOR

THREE STAGE BEVEL HELICAL GEARBOXES

Tran Thi Hong

1

, Tran Thi Phuong Thao

2

, Vu Ngoc Pi

2

, Le Hong Ky

3

, Nguyen Thi Hong Cam

2

,

Luu Anh Tung

2

and Le Xuan Hung

2

1

Nguyen Tat Thanh University, Ho Chi Minh City, Vietnam

2Thai Nguyen University of Technology, Thai Nguyen, Vietnam 3

Vinh Long University of Technology Education, Vinh Long, Vietnam E-Mail: [email protected]

ABSTRACT

This paper introduces a study on the calculation of optimum gear ratios of a three stage bevel helical gearbox. In this study, to find the optimum gear ratios, an optimization problem was performed. In the optimization problem, the gearbox length was chosen as the objective function. In addition, the effects of input parameters including the total gearbox ratio, the face width coefficients of the bevel and the helical gear sets, the allowable contact stress and the output torque were investigated. To evaluate the influence of these factors on the optimum gear ratios, a simulation experiment was designed and conducted by a computer program. Based on the results of the study, the influences of the input parameters on the optimum gear ratios were evaluated and models for calculation of the optimum gear ratios were found.

Keywords: gear ratio, optimum gearbox design, bevel helical gearbox.

1. INTRODUCTION

In optimum gearbox design, the optimum calculation of gear ratios is one of the most important missions. This is because the size, the mass as well as the cost of a gearbox depend powerfully on the gear ratios [1]. Consequently, optimum determination of gear ratios has been subjected to various researches.

Until now, there have been a variety of studies on the calculation of the gear ratios of different gearbox types. These studies have been carried out for helical gearboxes [1-9], for bevel helical gearboxes [1, 3, 10, 11] and for worm-gearboxes [3, 12-14]. These researches have been done on two-stage gearboxes [1-7, 12-14], three-stage gearboxes [8, 11] and four-three-stage gearboxes [9]. In addition, three main methods have been used to find the optimum gear ratios. These methods are the graph method [1, 2] (Figure-1), the practical method [3] and modeling method [4-13].

The gear ratios have been determined not only for a gearbox but also for a mechanical driven system. Recently, there have been several studies on the calculation of the optimum gear ratios for a mechanical driven system using a gearbox and a V-belt [15,16, 17] or a chain drive [18, 19].

Figure-1. The gear ratio of the bevel gear set versus the

This paper presents a study on the calculation of the optimum gear ratios of a three stage bevel helical gearbox with the objective as the minimum of the gearbox length. Also, the influence of the input factors on the optimum gear ratios were evaluated.

2. OPTIMIZATION PROBLEM

Figure-2. Calculation schema.

From Figure-2, the length of the gearbox is determined as (see Figure-2):

21

/ 2

2 3 w 23

/ 2

e w w

L

d

a

a

d

(1)

Where,

a

w2,

a

w3,

d

e21and

d

w23 are the center distance and the pitch diameters of the first and the third stage, respectively.

h

L

de21 dw13 dw23

dw12 dw22

aw2 aw3

de11

1

(2)

For the bevel gear set, the outer pitch diameter of the pinion is calculated by [20]:

2 11

2

/ 1

1

e e

d

 

R

u

(2)

In which,

R

eis the external cone distance of the

bevel gear set;

u

1 is the gear ratios the first stage. In addition, for the third helical gear stage,

d

w23 can be determined by the following equation [20]:

23 2 3 3/ 3 1

w w

d  au u

(3)

Where,

u

3 is the gear ratio of the third stage.

From equations (1), (2) and (3), it is noticed that for calculating the gearbox length L it is necessary to determine

R

e,

a

w2 and

a

w3.

2.1 Determining the external cone distance of the bevel gear set

The external cone distance

R

e is determined based on the pitting resistance of the straight bevel gear set [20]:

 

2

2 3

1

1

11 1

/ 1

1

e R H be be H

R

 

k

u

 

T k

k

  

k

u

(4)

Wherein,

k

R is the material coefficient; For steel

straight bevel gearings

k

R

50

(MPa1/3)[20];

k

be is the coefficient of the face width;

k

be

0.25

0.3

[20];

 

H1 is the allowable contact stress (MPa); KH1 is the

contact load ratio for pitting resistance of the bevel gear set. From the data in [20], the following regression equations were proposed for the calculation of KH1(with the coefficient of determination

R

2

1

):

2

1

0.25

0.2

1.02

H

K

 

k

 

k

(5)

Where,

k

k

be

u

1

/ 2

k

be

.

For the mechanic system including three gear sets, the following moment equilibrium equation can be found:

3

11 out

/

g bg hg b

T

T

u

  

(6)

In which, ug is the total ratio of the gearbox;

T

out

is the output torque (Nmm);

bgis the transmission

efficiency of the bevel gear;

hg

0.95

0.97

[20];

hg

is the transmission efficiency of the helical gear;

0.96

0.98

hg

[20];

bis the transmission efficiency of a pair of rolling bearing;

b

0.99

0.995

[20]. Choosing

bg0.96,

hg 0.97,

b

0.992

and substituting them into equation (6) gives:

11 1.101 out / g

T  T u (7)

Substituting

k

R

50

and (7) into (4) gets

 

2

2 3

1 1 1

51.6296

1

/ 1

e out H be be g H

R

u

 

T

k

k

   

k u u

(8)

Determining the center distance of the second stage

For the second stage, the center distance

a

w2is determined by [20]:

 

2

3

w2 m 2

1

12 H

/

H 2 ba2

a

k

u

 

T

k

 

u

(9)

Where, KH is the contact load ratio for pitting

resistance;kH 1.02 1.28 [20] and we can chose

1.1

H

k  ;

 

H is the allowable contact stress (MPa); In

practice,

 

H

360

420

(MPa);

k

m is the material

coefficient;

k

m=43 as the gear material is steel[20];

ba2 is the coefficient of wheel face width of the second stage of the gearbox;

ba2

0.3

0.35

[20];

For the mechanic system with two helical gear sets, the following equation can be found:

2 3

12 2 3

out hg b

T

T

 

  

u u

(10)

Where,

hgis the helical gear transmission

efficiency;

hg

0.96

0.98

[20];

bis the

transmission efficiency of a pair of rolling bearings;

0.99

0.995

be

[20]. Choosing

hg 0.97and

0.992

b

[20] gives:

12

1.0887

out

/

2 3

T

T

u u

(11)

Substituting (11) and kH 1.1 into (9) with

(3)

1

3

w2 2 2

2 2

45.6635 1 out

H g ba

T u

a u

u u

 

   

   (12)

The pitch diameter of the second stage then is calculated by [20]:

w 22

2

w2 2

/

2

1

d

 

a

u

u

(13)

2.2 Determining the center distance of the third stage

Calculating in the same way as in subsection 2.2, the center distance of the third stage

a

w3is determined by [20]:

 

13 3

w3 3 2

3 3

1

H

m

H ba

T

k

a

K

u

u

 

 

(14)

For the third stage the output torque is determined by:

2

13 3

out hg b

T

T

 

 

u

(15)

Choosing

hg 0.97and

b

0.992

as in subsection 2.2 gives:

12

1.0476

out

/

3

T

T

u

(16)

Substituting (16),

k

m=43 and kH 1.1 (as in section 2.2) into (14) gets:

 

3

w3 3 2 2

3 3

45.0814 1 out

H ba

T

a u

u

   

  (17)

The pitch diameter of the third stage then is calculated by [20]:

w 23

2

w3 3

/

3

1

d

 

a

u

u

(18)

Consequently, the optimization problem is defined as:

minimize

L

(19)

With the following constraints:

1

1

u

6

2

1

u

9

(20)

3

1

u

9

2.3 Experimental work

Table-1. Input factors.

Factor Code Unit Low High

Total gearbox ratio ug - 40 110

Coefficient of the face width of bevel gear set Kbe - 0.25 0.3

Coefficient of wheel face width of stage 2 xba2 - 0.3 0.35

Coefficient of wheel face width of stage 3 xba3 - 0.35 0.4

Allowable contact stress AS MPa 360 420

Output torque

T

out Nmm 10

5

(4)

Table-2. Experimental plans and output response.

Std Oder

Run Order

Center

Pt Blocks ug Kbe Xba2 Xba3

AS (MPa)

Tout

(Nm) u1 u2

1 1 1 1 40 0.25 0.3 0.35 360 100 1.52 5.97

22 2 1 1 110 0.25 0.35 0.35 420 100 3.11 7.11

59 3 1 1 40 0.3 0.3 0.4 420 10000 1.36 6.63

51 4 1 1 40 0.3 0.3 0.35 420 10000 1.53 6

58 5 1 1 110 0.25 0.3 0.4 420 10000 2.71 7.92

15 6 1 1 40 0.3 0.35 0.4 360 100 1.50 6.03

14 7 1 1 110 0.25 0.35 0.4 360 100 2.89 7.47

63 63 1 1 40 0.3 0.35 0.4 420 10000 1.50 6.03

61 64 1 1 40 0.25 0.35 0.4 420 10000 1.49 6

To inspect the effects of the input factors on the optimum gear ratios, a simulation experiment was designed and accomplished. For the experiment, a 2-level full factorial design was designated. Besides, 6 input factors were chosen for the discovering (Table-1). Thus,

the design was decided with

2

6

64

number of experiments. To accomplish the experiment, a computer program was created based on equations (19) and (20). The various levels of input factors and the output responses (the optimum gear ratios of the first and the second stages

u

1and

u

2) are described in Table-2.

3. RESULTS AND DISCUSSIONS

To evaluate the influence of input factors on the response and the relative strength of the influence, Figure-3 displays the main effect of each factor on the optimum gear ratios of the bevel gear set

u

1 (Figure-3a) and the second stage

u

2(Figure-3b). From the Figure, both partial gear ratios

u

1and

u

2grow considerably with the rise of the total ratio of the gearbox ug. In addition, they are also

affected by the wheel face width coefficients of the bevel gear and the helical gear sets (

k

ba,

ba1and

ba2). Additionally, the optimum gear ratios are not dependentonthe allowable contact stress and the output torque.

The Pareto chart of the standardized effects from the largest effect to the smallest effect is shown in Figure-4. It can be seen from the figure, that the bars symbolizing factors including the total gearbox ratio (factor A), the wheel face width coefficients of the bevel gear set (factor B) and the helical gear sets (factors C and D) and the interactions between them cross the reference line. Therefore, these factors are statistically significant at the 0.05 level with the response model.

The Normal Plot of the standardized effects for

1

u

(Figure-5a) and

u

2(5b) is presented in Figure-5. According to the figure, the total ratio of the gearbox

(factor A) is the most significant factor for both partial ratios

u

1and

u

2. It has a positive effect on both

u

1 and

u

2. In addition, the wheel face width coefficients of the bevel gear set

k

behave a small negative effect on

u

2and it has no effect on

u

1. Besides,

ba2(factor C) has a positive standardized effect on

u

1while it has a negative standardized effect on

u

2.Further more, the wheel face width coefficient of the third helical gear set

ba3(factor D) has a positive effect on

u

1and negative effect on

u

2.

Figure-6 shows the estimated effects and coefficients for the partial gear ratios of the bevel geat set

1

u

(Figure-6a) and the second helical gear set

u

2 (Figure-6b). It is revealed from this figure, the parameters which have a significant effect on a response with P-values lower than 0.05 are the total gearbox ratio

u

g, the wheel face

(5)

a)

[image:5.595.51.289.95.365.2]

b)

Figure 3. Main effects plot for

u

1 and

u

2 .

a)

b)

Figure-4. Pareto Chart of the Standardized Effects for

1

u

and

u

2 .

1 2 3

2 3

2 3 2 3

1.662 0.02658 0.264 0.76 5.464

0.011462 0.008064 0.016508

1.387 3.701 5.489

        

         

        

g be ba ba

g be g ba g ba

be ba be ba ba ba

u u k

u k u u

k k

 

 

    (21)

2 2

3 2

3 3 2 3

2.788 0.01843

3.182

7.318

31.63

0.02357

0.03643

0.04071

9

51

 

 

 

 

 

 

 

g be ba

ba g be g ba

g ba be ba ba ba

u

u

k

u

k

u

u

k

(22)

The equations (21) and (22) fit the data very well because all of the adj-R2 and pred-R2 are in high values (Figure-6). These equations are used to determine the optimum gear ratios of the first and the second stages

u

1

and

u

2. After that, the optimum gear ratio of the third

stage can be calculated by

u

3

u

g

/

u u

1

2

.

a)

[image:5.595.54.290.96.678.2]

b)

Figure-5. Normal Plot for

u

1and

u

2 .

4. CONCLUSIONS

[image:5.595.312.548.246.612.2]
(6)

a)

b)

Figure-6. Estimated Effects and Coefficients for

1

u

and

u

2

.

ACKNOWLEDGEMENTS

The work described in this paper was supported by Thai Nguyen University of Technology for a scientific project.

REFERENCES

[1] V.N. Kudreavtev; I.A. Gierzaves; E.G. Glukharev. 1971. Design and calculus of gearboxes (in Russian). Mashinostroenie Publishing, Sankt Petersburg.

[2] A.N. Petrovski, B.A. Sapiro, N.K. Saphonova. 1987. About optimal problem for multi-step gearboxes (in Russian). Vestnik Mashinostroenie. (10): 13-14.

[3] Romhild I., Linke H. 1992. Gezielte Auslegung Von Zahnradgetrieben mit minimaler Masse auf der Basis neuer Berechnungsverfahren. Konstruktion. 44: 229-236.

[4] Trinh Chat. 1996. Optimal calculation the total transmission ratio of helical gear units (in Vietnamese). Scientific Conference of Hanoi University of Technology. 74-79.

[5] G. Milou; G. Dobre; F. Visa; H. Vitila. 1996. Optimal Design of Two Step Gear Units, regarding the Main Parameters, VDI Berichte. (1230): 227-244.

[6] Vu Ngoc Pi. 2001. A method for optimal calculation of total transmission ratio of two step helical gearboxes. Procedings of the National conference on Engineering Mechanics, Ha Noi, October 12-13.

[7] Vu Ngoc Pi. 2008. A new study on optimal calculation of partial transmission ratios of two-step helical gearboxes, 2nd WSEAS Int. Conf on COMPUTER ENGINEERING and APPLICATIONS (CEA'08) Acapulco, Mexico, January 25-27.

[8] Vu Ngoc Pi. 2008. A new study on optimal calculation of partial transmission ratio of three-step helical reducers. The 3rd IASME / WSEAS International Conference on Continuum Mechanics, Cambridge, UK, February 23-25.

[9] Vu Ngoc Pi. 2008. Optimal Calculation of Partial Transmission Ratios of Four-Step Helical Gearboxes for Getting Minimal Gearbox Length, World Academy of Science. Engineering and Technology International Journal of Mechanical and Mechatronics Engineering. 2(1).

[10]Vu Ngoc Pi. 2000. A new and effective method for optimal calculation of total transmission ratio of two step bevel - helical gearboxes, International colloquium on Mechannics of Solids, Fluids, Structures & Interaction Nha Trang, Vietnam. 716-719.

[11]Vu Ngoc Pi, Nguyen Dang Binh, Vu Quy Dac, Phan Quang The. 2005. A new and effective method for optimal splitting of total transmission ratio of three step bevel-helical gearboxes. The Sixth Vietnam Conference on Automation, Hanoi. 175-180.

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reducers. Journal of Science and Technology, Thai Nguyen University. 4(36), (1): 70-73.

[13]Vu Ngoc Pi, Vu Quy Dac. 2007. Calculation of total transmission ratio of two step worm reducers for the best reasonable gearbox housing structure (in Vietnamese). Journal of Science and Technology, Thai Nguyen University. 1(41): 65-69.

[14]Vu Ngoc pi, Vu Quy Dac. 2007. Optimal calculation of partial transmission ratios of worm-helical gear reducers for minimal gearbox length. Journal of Science and Technology of 6 Engineering Universities. 61: 73-77.

[15]Vu Ngoc Pi, Tran Thi Phuong Thao, Le Thi Phuong Thao. 2015. A new study on optimum determination of partial transmission ratios of mechanical driven systems using a V-belt and two-step helical gearbox. Vietnam Mechanical Engineering Journal. (10): 123-125.

[16]Vu Ngoc Pi, Nguyen Thi Hong Cam, Nguyen Khac Tuan. 2016. Optimum calculation of partial transmission ratios of mechanical driven systems using a V-belt and two-step bevel helical gearbox. Journal of Environmental Science and Engineering A. 5: 566.

[17]Fujita, H., Cuong, N.D., Vu, N.P., Long, B.T., Puta, H. (Eds.). 2018. Determining optimal partial transmission ratios of mechanical driven systems using a V-belt drive and a helical reducer with second-step double gear-sets, Advances in Engineering Research and Application, Proceedings of the International Conference, ICERA 2018, Springer. pp. 261-269.

[18]Vu Ngoc Pi, Tran Thi Phuong Thao, Dang Anh Tuan. 2017. Optimum determination of partial transmission ratios of mechanical driven systems using a chain drive and two-step helical gearbox, Journal of Environmental Science and Engineering B. 6: 80.

[19]Fujita, H., Cuong, N.D., Vu, N.P., Long, B.T., Puta H. 2018. (Eds.), A study on determination of optimum partial transmis-sion ratios of mechanical driven systems using a chain drive and a three-step helical reducer, Advances in Engineering Research and Application, Proceedings of the International Conference, ICERA 2018, Springer. pp. 91-99.

Figure

Figure-4. Pareto Chart of the Standardized Effects for

References

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