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.