A new study on optimal calculation of partial transmission ... - wseas

2nd WSEAS Int. Conf on COMPUTER ENGINEERING and APPLICATIONS (CEA'08) Acapulco, Mexico, January 25-27, 2008

A new study on optimal calculation of partial transmission ratios of two-step helical gearboxes VU NGOC PI Faculty of Mechanical, Maritime and Materials Engineering Delft University of Technology Mekelweg 2, 2628 CD Delft THE NETHERLANDS [email protected] Abstract: - This paper introduces a new study on the applications of optimization and computer techniques for optimal calculation of the partial ratios of two-step helical gearboxes for minimal reducer length. In the paper, basing on moment equilibrium condition of a mechanic system including two gear units and their regular resistance condition, a new way for determining the partial ratios of two-step helical gearboxes were suggested. By using regression analysis, explicit models for calculation of the partial ratios are given. These models allow calculating the ratios accurately and simply. Key-Words: - Gearbox design; Optimal design; Helical reducer; Transmission ratio.

1 Introduction In gearbox design, it is known that the partial ratios are main factors affecting the size, dimension, mass, and the cost of the gearboxes. Therefore, optimal determination of the partial ratios of gearboxes has been subjected to many studies. Until now, there have been many studies on the prediction of helical gearboxes. For two-step helical gearboxes, the partial ratios can be determined by the following methods: -By graph method: in this method, the partial ratios are determined graphically. This method was used in several studies such as in studies of Kudreavtev V.N. [1] (see Figure 1) and Trinh Chat [2]. -By “practical method”: in this method, the optimal partial ratio is determined based on practical data. Using this method, G. Milou et al. [3] noted that the gearbox weight will be minimal if the ratio aw 2 / aw1 is from 1.4 to 1.6 ( aw1 , aw 2 are the center distances of the first and the second step, respectively). From this, the authors gave the optimal values of the partial ratios (in the table form). -By models: in this method, the partial ratios are determined by formulas which are found based on the results of optimization problem. This method has been used in order to get various objectives such as for getting the minimal gearbox mass [4] or for getting the minimal across section of the gearbox [5].

ISSN: 1790-5117

Fig. 1 Transmission ratios of steps 1 versus the total transmission ratio [1]

From previous researches, it is clear that there have been many studies on the splitting the total transmission ratio for two step helical gearboxes. However, the minimal length of the gearbox, an important objective, has not been carried out. This paper introduces a new result for optimal prediction of partial ratios for two-step helical gearboxes for getting the minimal gearbox length.

2 Determination of the length of the gearbox

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In practice, the length of a two step helical gearbox is decided by the length of L (see Fig. 2) which is determined by the following equation: L=

d w11 d + aw1 + aw 2 + w 22 2 2

(1)

The center distance of the first helical gear unit is calculated by:

ISBN: 978-960-6766-33-6


aw1 =

⎞ d w11 d w 21 d w 21 ⎛ d w11 + = ⋅⎜ + 1⎟ 2 2 2 ⎝ d w 21 ⎠

Using the design equation for pitting resistance of helical gear unit [6], the following equation is given for the first step:

Or aw1 =

⎞ d w 21 ⎛ 1 ⎜ + 1⎟ 2 ⎝ u1 ⎠

2 ⋅ T11 ⋅ K H 1 ⋅ u1 + 1

σ H 1 = Z M 1 ⋅ Z H 1 ⋅ Zε 1 ⋅

bw1 ⋅ d w211 ⋅ u1

≤ [σ H 1 ]

(5)

(2) From (5) we have:

[T11 ] =

bw1 ⋅ d w211 ⋅ u1 [σ H 1 ] 2 ⋅ ( u1 + 1) K H 1 ⋅ ( Z M 1 ⋅ Z H 1 ⋅ Zε 1 )2 2

(6)

In which, bw1 and dw11 are calculated as follows:

bw1 = ψ ba1 ⋅ aw1 = d w11 =

ψ ba1 ⋅ d w11 ⋅ ( u1 + 1) 2

d w 21 u1

(7)

(8)

Substituting (7) and (8) into (5) we get

[T11 ] =

ψ ba1 ⋅ d w3 21 ⋅ [ K 01 ]

(9)

4 ⋅ u12

Where

[ K01 ] = Fig. 2 Calculating schema for two-step helical gearbox

⎞ d w 22 ⎛ 1 ⎜ + 1⎟ 2 ⎝ u2 ⎠

⎞ d ⎛ 1 ⎞ d w 21 ⎛ 2 ⋅ ⎜ + 1⎟ + w 22 ⋅ ⎜ + 2 ⎟ 2 ⎝ u1 2 ⎝ u2 ⎠ ⎠

(10)

1/ 3

(11)

(3)

Calculating in the same manner, we have the following equation for the second gear unit: ⎛ 4 [T12 ] u22 ⎞ =⎜ ⎜ ψ ba 2 [ K 02 ] ⎟⎟ ⎝ ⎠

1/ 3

d w 22 (4)

In the above equations, u1, u2 are transmission ratios of helical gear units 1 and 2; dw11, dw21, dw12, dw22 are pitch diameters (mm) of the gear units 1 and 2; aw1, aw2 are center distances (mm) of the gear units 1 and 2, respectively.

ISSN: 1790-5117

2

⎛ 4 [T11 ] u12 ⎞ d w 21 = ⎜ ⎜ ψ ba1 [ K 01 ] ⎟⎟ ⎝ ⎠

Substituting (2) and (3) into (1), Equation 1 can be rewritten as follows: L=

K H 1 ⋅ ( Z M 1 ⋅ Z H 1 ⋅ Zε 1 )

From (9) the pitch diameter dw21 can be calculated by

Using the same way for the second step we have: aw 2 =

[σ H 1 ]2

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(12)

In the above equations, Z M 1 , Z H 1 , Z ε 1 are coefficients which consider the effects of the gear material, contact surface shape, and contact ratio of the first gear unit when calculate the pitting resistance; [σ H 1 ] is allowable contact stresses of bevel gear unit; ψ ba1 and ψ ba 2 are coefficients of helical gear face width of steps 1and 2, respectively. From the condition of the moment equilibrium of the ISBN: 978-960-6766-33-6


Where, K c = ⎡⎣K 02 ⎤⎦ / ⎡⎣K 01 ⎤⎦ .

first step and the regular resistance condition of the system we have:

3 Optimization problem and results

Tr [Tr ] 2 = = u1 ⋅ u2 ⋅ηbrt ⋅ηo2 T11 [T11 ]

From Equation 19, the optimal problem for finding the minimal length of the gearbox can be expressed as follows: Objective function:

(13)

Where, ηbrt is helical gear transmission efficiency (ηbrt is from 0.96 to 0.98 [6]); ηo is transmission

min L = f (uh ; u2 )

efficiency of a pair of rolling bearing (ηo is from 0.99 to 0.995 [6]). Choosingη brt = 0.97 , η o = 0.992 and substituting them into (15) we have

[T ] [T11 ] = 0.9259r⋅ u

1

With the following constraints:

u h min ≤ u h ≤ u h max u 2 min ≤ u 2 ≤ u 2 max

(14)

⋅ u2

K c min ≤ K c ≤ K c max

Substituting (14) into (11) we have 1/ 3

(15)

A computer program was built for performing the above optimization problem. The data used in the optimization program was chosen: K c is from 1 to 1.3, ψ ba1 and ψ ba 2 is from 0.25 to 0.4 [6], u2 is from 1 to 9 [1]; uh is from 5 to 40. Based on the results of the optimization program, the following regression model was found for calculation the optimal values of partial ratio of the second step u2:

For the second step we have:

[T ] Tr = r = u2 ⋅ηbrt ⋅ηo T12 [T12 ]

(16)

With η brt = 0.97 and η o = 0.992 Equation 16 becomes

(see above)

[Tr ]

We now substitute (17) into (12) to obtain d w 22 :

d w 22

1/ 3

⎞ ⎟⎟ ⎠

(18)

Substituting (15) and (18) into (4) with the note that u1 = uh / u2 we have 1 ⎛ [T ] ⎞ L= ⎜ r ⎟ 2 ⎜⎝ [ K 01 ] ⎟⎠

1/ 3

⎡⎛ 4.3201 ⋅ u h ⎢⎜ ⎢⎜⎝ ψ ba1 ⋅ u22 ⎣

1/ 3

⎛ 4.1571 ⋅ u2 ⎞ +⎜ ⎟ ⎝ ψ ba 2 ⋅ K c ⎠

ψ ⎞ ⎛ u2 ≈ 1.1966 ⋅ ⎜ K c ⋅ ba 2 ⎟ ψ ba1 ⎠ ⎝

(17)

0.9622 ⋅ u2

⎛ 4.1571 ⋅ [Tr ] ⋅ u2 =⎜ ⎜ ψ ⋅[K ] 02 ba 2 ⎝

1/ 3

⎞ ⎟⎟ ⎠

⎛2 ⎞ ⋅ ⎜ + 1⎟ + ⎝ u1 ⎠

(19)

0.35

⋅ uh0.2939

(22)

The regression models fit very well with the data (with the coefficients of determination 2 was R = 0.9994 ). Equation 22 is used to calculate the transmission ratios u2 of the second step. After determining u2, the transmission ratio of the first step u1 can be determined as follows: u1 =

uh u2

(23)

4 Conclusion

⎛ 1 ⎞⎤ ⋅ ⎜ + 2 ⎟⎥ ⎝ u2 ⎠ ⎥⎦

ISSN: 1790-5117

(21)

ψ ba 2 min ≤ ψ ba 2 ≤ ψ ba 2 max

⎛ 4.3201 ⋅ [Tr ] ⋅ u1 ⎞ d w 21 = ⎜ ⎜ ψ ⋅ [ K ] ⋅ u ⋅ u ⎟⎟ 01 1 2 ⎠ ⎝ ba1

[T12 ] =

(20)

It can be concluded that the minimal length of two-step helical gearboxes can be obtained by optimal splitting the total transmission ratio of the gearboxes. Models for prediction of the optimal partial ratios

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ISBN: 978-960-6766-33-6


for getting the minimal length of the gearboxes have been proposed. By introducing explicit models, the partial ratios of the gearboxes can be calculated accurately and simply.

References: [1] V.N. Kudreavtev; I.A. Gierzaves; E.G. Glukharev, Design and calculus of gearboxes (in Russian), Mashinostroenie Publishing, Sankt Petersburg, 1971, pp. 190-193. [2] Trinh Chat, Optimal calculation the total transmission ratio of helical gear units (in Vietnamese), Scientific Conference of Hanoi University of Technolog (1996) 74-79. [3] G. Milou; G. Dobre; F. Visa; H. Vitila, Optimal Design of Two Step Gear Units, regarding the Main Parameters, VDI Berichte No 1230, 1996, pp. 227-244. [4] Romhild I. , Linke H., Gezielte Auslegung Von Zahnradgetrieben mit minimaler Masse auf der Basis neuer Berechnungsverfahren, Konstruktion 44 (1992) 229- 236. [5] Vu Ngoc Pi, A new method for optimal calculation of total transmission ratio of two step helical gearboxes, The Nation Conference on Engineering Mechanics, Hanoi, October 12-13, 2001, 133- 136. [6] Trinh Chat, Le Van Uyen, Design and calculus of Mechanical Transmissions (in Vietnamese), Educational Republishing House, Hanoi, 1998, pp. 90-145.

ISSN: 1790-2769

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ISBN: 978-960-6766-32-9