Integrated optimization method for helical gear hobbing parameters considering machining efficiency, cost and precision


The formulation of dry hobbing processing parameters depends heavily on mass experiments and the experiences of skilled technicians, and the use of unsuitable parameters will lead to high cost, low efficiency and severe precision defects. To resolve this issue, a helical gear processing parameter optimization method (PPOM-HG) is proposed in this paper. First, an efficiency-cost-accuracy triple-target optimization model is established. A manufacturing efficiency model is established through a detailed analysis of the geometric trajectory of a hob. A helical gear manufacturing cost model is established by analyzing the power curve of the hobbing machine and the hob’s lifetimes under various processing parameters. A modified correlation analysis random forest (CARF) model is designed for predicting gear machining precision, which replaces the traditional empirical precision function. Then, for searching for the optimal solution of the established efficiency-cost-precision triple-target model, an adaptive multiobjective fusion evolutionary algorithm (AMFEA) with adaptive evolution parameters is proposed. Finally, via many helical gear machining experiments, the validity and advantages of the proposed CARF and AMFEA methods are demonstrated, and the selection strategy of the Pareto front solution under various conditions is discussed.

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Data availability

The raw/processed data we collected for the helical gear processing parameters optimization research cannot be shared at this time as the data also forms part of an ongoing study.

Code availability

The method codes involved in this paper are available.


α h :

Helix angle of hob cutter

α i a :

Installation angle of hob cutter

β n :

Helix angle of gear

η,λ,ρ :

Power factor of hob spindle speed, feed rate and cutting depth respectively

a t :

Cutting depth/hight of gear tooth

A p c :

Adaptive adjust crossover probability

A p m :

Adaptive adjust mutation probability

B n :

Width of helical gear workpiece

C e l e :

Energy cost helical gear machining

C H :

Hardness of hobbing cutter coating

C m a x :

Random neighborhood parameter lower bound

C m a x :

Random neighborhood parameter upper bound

C m :

Random neighborhood parameter

C S U M :

Helical gear machining cost

\(C_{tool}^{k}\) :

Hob cutter wear loss cost

D e n t :

Stroke of hob-shaft entry/retract stage

d g :

Base circle diameter of gear

d H :

Running stroke of cutting process

d i d l e :

Idle stroke in cutting process

d i n :

Hob cutting-in stroke

d n :

Addendum circle diameter of gear

d o u t :

Hob cutting-out stroke

D t :

Diameter of hobbing cutter

F b t :

Price of uncoated hob

\(F_{coating}^{k}\) :

Price of coating

F e l e :

Unit price of electricity charge

f f α :

Gear tooth profile form deviation

f f β :

Gear helix form deviation

F r :

Radial run-out

I t e r :

Current iteration number

M n :

Normal module of gear

M a x G e n :

The maximum number of iterations limitation

M e :

Over pin measurement

\(N_{k}^{(R)}\) :

Rated machining number for specific hob

N s :

Number of hobbing cutter heads

p DE :

Selection probability of moDE operator

p GA :

Selection probability of moGA operator

P c u t t i n g :

Power of cutting process

P e n t :

Power of hob-shaft entry process

P I N :

Inherent parameters of helical gear

P m s :

Mutation selection probability operator

P M T :

Machining parameters of machine tool

P r e t :

Power of hob-shaft retract process

\(P_{spindle}^{(cut)}\) :

Cutter spindle power in cutting process

\(P_{spindle}^{(ent)}\) :

Cutter spindle power in entry process

P s t a n d b y :

Power of standby process

p c l o :

Lower bound of adaptive adjust crossover operator

p c u p :

Upper bound of adaptive adjust crossover operator

pm :

Mutation probability

P s i z e :

Population size

Q E S :

Helical gear machining quality inspection item

Q M S t :

Qualified measurement size of t th inspection index

R a :

Gear surface roughness

S h o b :

Energy consumption of one helical gear hobbing

T c t :

Time period of changing hob

t c u t t i n g :

Time period of cutting process

t e n t :

Time period of cutter-shaft entry

t i n :

Time period of din stroke

T m :

Time period of machining a single helical gear

t o u t :

Time period of dout stroke

t r e t :

Time period of cutter-shaft retract

t s b :

Time period of standby process

\(Tor_{t}^{Max}\) :

Maximum allowable tolerance of t th quality inspection index

\(V_{cutting}^{(re)}\) :

Recommended cutting speed for specific hob

V e n t :

Moving speed of hob-shaft entry/retract stage

\(V_{feed}^{(re)}\) :

Recommended feed rate for specific hob

V f R n :

Rated feed rate of hob

V f :

Feed rate of hob cutter

V s R n :

Rated cutting speed of hob

V s :

Rotate speed of hobbing machine cutter spindle

V w o r k t a b l e :

Worktable shaft turning speed

V w :

Cooling wind speed

W :

Total data set

w c :

Weight of manufacturing cost criterion

w e :

Weight of manufacturing efficiency criterion

w p :

Weight of machining precision criterion

\(x_{i}^{Ind}\) :

Floating encoding individual of AMFEA

x g :

Modification coefficient of gear

X m a n :

Manufacturing parameters of helical gear

\(y_{n}^{pre}\) :

Prediction of machining precision

Z n :

Number of gear teeth

\({\bar {P}_{io}}\) :

Average power of cutting-in and cutting-out process


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This research is supported by the “Chongqing Technology Innovation and Application Development Special Project (cstc2019jscx-mbdxX0016)”, and “Basic Scientific Research Business Expenses of Central Universities of Chongqing University (2019CGCG0003, 2019CGCG0004, 2019CDCGJX315)”.

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Correspondence to Ping Yan.

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Conflict of interest

We would like to submit the enclosed manuscript entitled “Integrated optimization method for processing parameters of helical gear considering machining efficiency, cost and precision”, which we wish to be considered for publication in The International Journal of Advanced Manufacturing Technology. No conflict of interest exits in the submission of this manuscript, and the manuscript is approved by all authors for publication. I would like to declare on behalf of my co-authors that the work described is original research that has not been published previously, and not under consideration for publication elsewhere, in whole or in part. All the authors listed have approved the manuscript that is enclosed.

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Authors’ contributions

Dayuan Wu contributed to the conception of the study and wrote the manuscript; Ping Yan helped perform the analysis with constructive discussions. You Guo performed the gear machining experiment and contributed to data preprocessing; Han Zhou contributed to data preprocessing and analysis; Runzhong Yi performed the gear machining experiment.

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Wu, D., Yan, P., Guo, Y. et al. Integrated optimization method for helical gear hobbing parameters considering machining efficiency, cost and precision. Int J Adv Manuf Technol (2021).

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  • Helical gear
  • Multiobjective optimization
  • Processing parameter
  • Precision prediction
  • Cost modeling
  • Evolutionary algorithm