//Logo Image
Author:Yeh-Liang Hsu, Shu-Gen Wang (2000-02-18); modified: Yeh-Liang Hsu (2003-02-07); recommended: Yeh-Liang Hsu (2000-04-17).
Note: This paper is Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, Vol. 216, No. 4, 2002, p 565~569.

Minimizing Angular Backlash of a Multi-Stage Gear Train

Abstract

An optimization model is constructed to select the optimum reduction ratios that minimize the total angular backlash of a gear train, under constraints on total reduction ratio and available space. It is found that a proper layout of reduction ratios has major effect on the total angular backlash of a gear train.

Keywords: gear train, angular backlash, optimization.

Introduction

The prime function of gear transmission systems in many precision machines is to accurately transmit angular displacement. In such applications, gear backlash causes an unexpected angular position error when a rotating gear train reverses, and it also induces a transient impact force on the mating gear surface due to the moment of inertia of the system.

There is enormous amount of research literatures addressing precision gear design. Various design requirements have been considered, in particular, minimizing the transmission error has received many attentions. Yoon and Rao[1996] minimized the static transmission error using cubic splines for gear tooth profile. Iwase and Miyasaka[1996] modified tooth profile to minimize transmission error of helical gears, while Shibata et al.[1997] tried to find the optimum tooth profile that minimizes transmission error for hypoid gears. Using quadrature factorial models, Yu and Ishii[1998] modified the tooth profile to find a “robust optimum” that has minimum expected transmission error.

However, modifying the tooth profile in order to minimize the transmission error may not be a practical option for many precision machine designers. Typically, design decisions faced by the designers are how to select proper reduction ratios, gear quality grades and tolerances, so that the required transmission precision of a gear transmission system can be obtained in a most cost-effective way.

In this research, it is found that a proper layout of reduction ratios of a multi-stage gear train has major effect on its total angular backlash. An optimization model is constructed to find the optimum reduction ratios that minimize the total angular backlash of a gear train, under constraints on total reduction ratio and available space.

Angular backlash of a gear train

Factors contributing to angular backlash mainly come from two sources: (1) original manufacturing errors and (2) assembly errors, such as center distance tolerance, bearing tolerance, shaft and bearing tolerance, etc. In this study, the original manufacturing errors and the center distance tolerance are considered.

The original manufacturing errors of a gear is specified by its quality grade. The quality grades of spur gears and helical gears range from 0 to 9 in JIS B 1703, and ranges from 0 to 12 in ISO 1328. These two standards also have slightly different definitions in error estimation for each quality grade. According to JIS B 1703 [1995], the linear backlash at the pitch line d is estimated by

                                                                             (1)

where r and m are the radius and module of the gear. The quality grade of the gear is reflected in the coefficient B in Eq.(1). For example, the value B ranges from 10 to 25 for grade 0 gears (the finest quality), from 10 to 28 for grade 1 gears, and from 10 to 90 for grade 8 gears.

Figure 1 shows the layout of a typical three-stage gear train. The reduction ratios of the three gear pairs are , , and . The angular backlash of gear i can be expressed as

                                                                                                     (2)

where  is the linear backlash, and  is the radius of gear i. The total angular backlash can be expressed as a linear sum of backlashes of individual gears, reflected at the output shaft of the gear train [Chironis, 1971]. For the gear train in Figure 1, the total angular backlash due to original manufacturing errors reflected at the shaft of gear 6 can be expressed as,

                                                     (3)

Note that , , and .

Figure1. A three-stage gear transmission system

Considering the center distances tolerance, as shown in Figure 2, the effect of a radial separation on the linear backlash is

                                                                                                (4)

where f and C are the pressure angle and the center distance tolerance, respectively. Similar to the derivation of Eq.(3), the total angular backlash due to the center distance tolerance reflected at the shaft of gear 6 can be expressed as,

                                                   (5)

Figure 2. The influence of center distance tolerance to gear backlash

It will be cumbersome to consider all possible combinations of signs (plus or minus) of individual tolerances. Therefore the total angular backlash at the output shaft of a gear train  is expressed as the geometric sum of individual tolerances  and ,

                                                                                                                                         (6)

The optimum design model

An optimum design model for a multi-stage gear train is constructed in this section. The objective function is to minimize the total angular backlash of the gear train (Eq.(6)). There are two constraints in this model: the total available space and the total reduction ratio of the gear train.

Referring to Figure 1, the allowable space occupied by the gear train is limited within a given value W,

                                                                                            (7)

The total reduction ratio of the gear train has to be larger than Kt, therefore,

                                                                                                    (8)

Finally the optimum design model can be written as

Minimize

subject to       

                       

                       

                       

                       

                       

                       

                       

                       

                       

                       

                       

                         

                       

                                                                                  (9)

The last two constraints in Eq.(9) post upper and lower bounds to the design variables ri and ki, where Nmin is the minimum number of teeth of the gears, and Kmax is the maximum allowable reduction ratio. Note that W, Kt, B, C, M, Nmin, and Kmax are written in capital letters to represent parameters that are given for a certain case, but may vary from case to case.

Solving the optimum design model

To solve for numerical values of this optimization model, we first assume a set of parameters: W=100mm, Kt=120, B=30, C=0.020mm, Nmin=18, Kmax=7, and M=0.5mm. In this study, numerical optimization software GAMS[Brooke, Kendrick, and Meeraus, 1992] was used to solve for the optimum reduction ratios that minimize the total angular backlash of the gear train (Table 1). Since the number of teeth has to be an integer, the radii of the gears are rounded up to the closest integers multiplied by M/2. Both continuous and integer solutions are presented in the table. Note that with the integer solution, some of the constraints in Eq.(9) may be slightly violated.

Table 1. Optimum reduction ratios that minimize total angular backlash

 

continuous sol.

7.18

3.53

4.85

7.00

11.03

38.97

4.50

21.83

5.21

36.47

integer sol.

7.20

3.55

4.83

6.95

11.00

39.00

4.50

21.75

5.25

36.50

Table 2 shows the reduction ratios that “maximize” the total angular backlash of the gear train under the same requirements on space and total reduction ratio. Comparing Table 1 and 2, we can see that by simply varying the reduction ratios, the maximum total angular backlash is 2.52 times the minimum total angular backlash in this example. Note that both cases have the same quality grade gears and the same center distance tolerances, therefore their manufacturing costs should be about the same.

Table 2. Reduction ratios that maximize total angular backlash

 

continuous sol.

18.08

7.00

7.00

2.45

4.50

31.50

4.50

31.50

4.50

11.02

integer sol.

18.12

7.00

7.00

2.44

4.50

31.50

4.50

31.50

4.50

11.00

Also note that  for the reduction ratios that minimize the total angular backlash. On the contrary,  for the reduction ratios that maximize the total angular backlash. This can be easily explained by observing the objective function in Eq.(10). The reduction ratio  appears in the denominators of all 6 terms of the objective function, while  appears in the denominators of 4 terms, and  appears in the denominators of only 2 terms. To minimize the total angular backlash, obviously  has the top priority to be as large as possible. The reduction ratio  has the second priority, followed by .

At the optimum solution shown in Table 1, space constraints and  are active. The total angular backlash can be further reduced if the space constraints are relaxed. As shown in Figure 3, for the same gear train discussed above, the minimum total angular backlash decreases as W increases.

Figure 3. Minimum total angular backlash vs. change of available space

Figure 4 and 5 shows the change of minimum total angular backlash with respect to the change of parameters B and C. Comparing Figure 4 and 5, we can see that the total angular backlash is more sensitive to the parameter B in this case. From this kind of parameter analysis, designers can evaluate the cost of changing into higher quality gears and the cost of tightening the center distance tolerance, to decide how to reduce the total angular backlash in a most cost-effective way.

Figure 4. Minimum total angular backlash vs. the change of parameter B on gear quality

Figure 5. Minimum total angular backlash vs. the change of center distance tolerance

Conclusions

This paper presents an optimization model for finding the optimum reduction ratios that minimize the total angular backlash of a gear train. While the estimation of total angular backlash in the optimization model may not be quantitatively exact, several qualitative conclusions can be drawn:

l          Under the same design requirements on space and total reduction ratio, simply varying the reduction ratios can reduce the total angular backlash of a gear train, while the manufacturing cost stays the same.

l          To minimize the total angular backlash of a gear train, the reduction ratio closer to the output shaft has higher priority to be made as large as possible.

l          The total angular backlash can be further reduced if the constraints on available space are relaxed.

This optimization model and the parameter analysis also provide a means to evaluate how to reduce the total angular backlash of a gear train in a most cost-effective way.

Reference

Brooke, Kendrick, and Meeraus, 1992. GAMS, A User's Guide, The Scientific Press, South San Francisco, California.

Chironis, N.P. ed., 1971. Gear Design and Application, McGraw-Hill, New York, pp. 236.

Iwase, Y., Miyasaka, K., 1996. “Proposal of modified tooth surface with minimized transmission error of helical gears,” JSAE Review, Vol. 17, No. 2, pp. 191-193.

JIS B1703, “Backlash for spur and helical gears,” 1995. JIS Handbook, Machine Elements, Japanese Standards Association, pp. 1689-1725.

Shibata, Y., Kondou, N., Ito, T., 1997. “Optimum tooth profile design for hypoid gear,” JSAE Review, Vol. 18, No. 3, pp. 283-287.

Yoon, K,Y., and Rao, S.S., 1996. “Dynamic load analysis of spur gears using a new tooth profile,” Journal of Mechanical Design, Vol. 118, No. 1, pp. 1-6.

Yu, J.-C., Ishii, K., 1998. “Design optimization for robustness using quadrature factorial models,” Engineering Optimization, Vol. 30, No. 3/4, pp. 203-225.