Advertisement

Optimum design of straight bevel gears pair using evolutionary algorithms

  • Abolfazl Zolfaghari
  • Masoud Goharimanesh
  • Ali Akbar AkbariEmail author
Technical Paper

Abstract

Straight bevel gear is a type of gear which is widely used in mechanical systems to transmit power between perpendicular rotating axes. Designing straight bevel gears with the least possible volume is of great importance in industry since it results in a decrease in energy consumption and the material requirement in manufacturing. In this paper, employing two powerful optimization algorithms, simulated annealing algorithm (SA) and genetic algorithm (GA), techniques for advanced optimization, coupled with American Gear Manufacturers Association (AGMA) instructions the volume of straight bevel gears pair is minimized and the corresponding design variables are obtained. These variables include majors, including teeth number, module and face width. Using a traditional technique, recommended values of the design variables by AGMA, a design example was performed and the values were obtained. Then, the suggested techniques were utilized to get the values. The comparison between the results of all techniques shows that proposed optimization algorithms are considerably capable of minimizing the volume. It indicates that improvement in the attained volume varies between 1.56 and 17.40% for SA and 9.28 and 23.15% for GA.

Keywords

Straight bevel gear Optimum design Simulated annealing algorithm Genetic algorithm AGMA 

List of symbols

AiG

Gear inner cone distance (mm)

Am

Mean cone distance (mm)

b

Face width (mm)

bilp

Pinion limit inner dedendum (mm)

bip

Pinion inner dedendum (mm)

bp

Pinion mean dedendum (mm)

bmin/bmax

Lower/upper limit range of face width (mm)

d

Pitch diameter (mm)

e

Base of natural logarithm

Ei/Ei−1

Energy level of the system at current/previous position

f(X)

Equality constraint

F(X)

Objective function

g(X)

Inequality constraint

i

Number of steps in searching procedure

KA

Overload factor

KV

Dynamic factor

K

Load distribution factor

Kθ

Temperature factor

k

Boltzmann constant

m

Module (mm)

met

Outer transverse module (mm)

n1

Input speed (rpm)

Pi

Acceptance probability of the found solution

Pr

A random generated number (0 ≤ P r ≤ 1)

Qv

Transmission accuracy number

r

Cone pitch (mm)

SH/SF

Safety factor for contact/bending stress

Ti/T(i−1)

System temperature of the system at current/previous position

Vstraight bevel gear

Volume of straight bevel gear (mm3)

Vol

Total volume of pinion and gear

Wt

Transmitted load (N)

X

Vector of design variables

YNT

Stress-cycle factor for bending strength

Yx

Reliability factor for bending strength

Yz

Size factor for bending

Yβ

Length-wise curvature factor for bending strength

z

Teeth number

z1/z2

Teeth number of pinion/gear

ZE

Elastic coefficient for pitting resistance ([N/mm2]0.5)

ZI

Contact geometry factor

ZJ

Bending geometry factor

ZNT

Stress-cycle factor for pitting resistance

Zx

Size factor for pitting resistance

Zxc

Crowning factor for pitting

Zw

Hardness-ratio factor

Zz

Reliability factor for pitting

α

Cooling rate of the system

γ

Pinion pitch angle (rad)

λ

Cone angle (rad)

σHlim/σFlim

Allowable contact/bending stress number (N/mm2)

δ1/δ2

Cone angle of pinion/gear (rad)

δG/δP

Gear/pinion dedendum angle (rad)

ΨiG

Inner gear spiral angle (rad)

φ

Normal pressure angle at pitch surface (rad)

φTi

Inner transverse pressure angle (rad)

Abbreviations

AGMA

American Gear Manufacturers Association

GA

Genetic algorithm

min

Minimum

N

No

SA

Simulated annealing

TA

Traditional algorithm

Y

Yes

References

  1. 1.
    Radzevich SP (2012) Dudley’s handbook of practical gear design and manufacture, 2nd edn. CRC Press, Boca RatonCrossRefGoogle Scholar
  2. 2.
    Budynas RG, Nesbit JK (2011) Shigley’s mechanical engineering design, 9th edn. McGraw-Hill, New York [Part 13 (624-732) and part 15 (786-824)] Google Scholar
  3. 3.
    Vahabi H, Panahi MS, Shirazinezhad RP, Imenabadi A (2015) A neuro-genetic approach to the optimal design of gear-blank lightening holes. J Braz Soc Mech Sci Eng. doi: 10.1007/s40430-015-0362-0 Google Scholar
  4. 4.
    Wang H, Wang HP (1994) Optimal engineering design of spur gear set. Mech Mach Theory 29:1071–1080. doi: 10.1016/0094-114X(94)90074-4 CrossRefGoogle Scholar
  5. 5.
    Thompson DF, Gupta S, Shukla A (2000) Tradeoff analysis in minimum volume design of multi-stage spur gear reduction units. Mech Mach Theory 35:609–627. doi: 10.1016/S0094-114X(99)00036-1 CrossRefzbMATHGoogle Scholar
  6. 6.
    Huang H, Tian Z, Zuo MJ (2005) Multiobjective optimization of three-stage spur gear reduction units using interactive physical programming. J Mech Sci Technol 19(5):1080–1086. doi: 10.1007/BF02984029 CrossRefGoogle Scholar
  7. 7.
    Marjanovic N, Isailovic B, Marjanovic V, Milojevic Z, Blagojevic M, Bojic M (2012) A practical approach to the optimization of gear trains with spur gears. Mech Mach Theory 53:1–16. doi: 10.1016/j.mechmachtheory.2012.02.004 CrossRefGoogle Scholar
  8. 8.
    Golabi S, Fesharaki JJ, Yazdipoor M (2014) Gear train optimization based on minimum volume/weight design. Mech Mach Theory 73:197–217. doi: 10.1016/j.mechmachtheory.2013.11.002 CrossRefGoogle Scholar
  9. 9.
    Tudose L, Buiga O, Stefanache C, Sobester A (2010) Automated optimal design of a two-stage helical gear reducer. Struct Multidisc O 42:429–435. doi: 10.1007/s00158-010-0504-z CrossRefGoogle Scholar
  10. 10.
    Padmanabhan S, Raman VS, Chandrasekaran M (2014) Optimisation of gear reducer using evolutionary algorithm. Mater Res Innov 18(6):378–382. doi: 10.1179/1432891714Z.000000000983 Google Scholar
  11. 11.
    Yang X (2008) Introduction to mathematical optimization—from linear programming to mataheuristics. Cambridge International Science Publishing, CambridgeGoogle Scholar
  12. 12.
    Zhao S, Li J, Zhang C, Zhang W, Lin X, He X, Yao Y (2015) Thermo-structural optimization of integrated thermal protection panels with one-layer and two-layer corrugated cores based on simulated annealing algorithm. Struct Multidisc O 51:479–494. doi: 10.1007/s00158-014-1137-4 CrossRefGoogle Scholar
  13. 13.
    Mendi F, Baskal T, Boran K, Boran FE (2010) Optimization of module, shaft diameter and rolling bearing for spur gear through genetic algorithm. Expert Syst Appl 37:8058–8064. doi: 10.1016/j.eswa.2010.05.082 CrossRefGoogle Scholar
  14. 14.
    Bangert P (2012) Optimization for industrial problems. Springer, Berlin doi:  10.1007/978-3-642-24974-7 CrossRefzbMATHGoogle Scholar
  15. 15.
    Yokota T, Taguchi T, Gen M (1998) A solution method for optimal weight design problem of the gear using genetic algorithms. Comput Ind Eng 35:523–526. doi: 10.1016/S0360-8352(98)00149-1 CrossRefGoogle Scholar
  16. 16.
    Marcelin JL (2001) Genetic optimization of gears. Int J Adv Manuf Tech 19:910–915. doi: 10.1007/s001700170101 CrossRefGoogle Scholar
  17. 17.
    Fonseca DJ, Shishoo S, Lim TC, Chen DS (2005) A genetic algorithm approach to minimize transmission error of automotive spur gear sets. 19(2):153–179. doi: 10.1080/08839510590901903 Google Scholar
  18. 18.
    Gologlu C, Zeyveli M (2009) A genetic approach to automate preliminary design of gear drives. Comput Ind Eng 57:1043–1051. doi: 10.1016/j.cie.2009.04.006 CrossRefGoogle Scholar
  19. 19.
    Zhang X, Rong Y, Yu J, Zhang L, Cui L (2011) Development of optimization design software for bevel gear based on integer serial number encoding genetic algorithm. JSW 6:915–922. doi: 10.4304/jsw.6.5.915-922 Google Scholar
  20. 20.
    Dhafer G, Jérôme B, Philippe V, Michel O, Mohamed H (2012) Robust optimization of gear tooth modifications using a Genetic Algorithm. In: Haddar M (ed) Design and modeling of mechanical systems. Springer, Heidelberg, pp 189–197Google Scholar
  21. 21.
    Blanco JC, Gobbi M, Muñoz LE, (2014). Gear train optimization of a hybrid electric off-road vehicle. In: International design engineering technical conferences and computers and information in engineering conference, Buffalo, New York, USA, 1–9Google Scholar
  22. 22.
    Tudose L, Buiga O (2014) Optimal mass minimization design of a two-stage coaxial helical speed reducer with Genetic Algorithms. Adv Eng Soft 68:25–32. doi: 10.1016/j.advengsoft.2013.11.002 CrossRefGoogle Scholar
  23. 23.
    Savsani V, Rao RV, Vakharia DP (2011) Optimal weight design of a gear train using particle swarm optimization and simulated annealing algorithms. Mech Mach Theory 45:531–541. doi: 10.1016/j.mechmachtheory.2009.10.010 CrossRefzbMATHGoogle Scholar
  24. 24.
    Shi Y, Eberhart R (1998) A modified particle swarm optimizer. In: IEEE world congress computational intelligence, Anchorage, Alaska, pp 69–73Google Scholar
  25. 25.
    Shieh H, Kuo C (2011) Modified particle swarm optimization algorithm with simulated annealing behavior and its numerical verification. Appl Math Comput 218:4365–4383. doi: 10.1016/j.amc.2011.10.012 zbMATHGoogle Scholar
  26. 26.
    Zhao S, Li J, Zhang C, Zhang W, Lin X, He X, Yao Y (2015) Thermo-structural optimization of integrated thermal protection panels with one-layer and two-layer corrugated cores based on simulated annealing algorithm. Struct Multidisc Optim 51:479–494. doi: 10.1007/s00158-014-1137-4 CrossRefGoogle Scholar
  27. 27.
    Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220(4598):671–680MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Correia DS, Gonçalves CV, Sebastião S, Junior C, Ferraresi VA (2004) GMAW welding optimization using genetic algorithms. J Braz Soc Mech Sci Eng. doi: 10.1590/S1678-58782004000100005 Google Scholar
  29. 29.
    Thiagarajan C, Sivaramakrishnan R, Somasundaram S (2012) Modeling and optimization of cylindrical grinding of Al/SiC composites using genetic algorithms. J Braz Soc Mech Sci Eng 34(1):32–40Google Scholar
  30. 30.
    ANSI/AGMA 2003-B97 (2003) Design Manual for Bevel Gears. Alexandria, Virginia. USA: American Gear Manufacturers Association (AGMA), 1–32Google Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2017

Authors and Affiliations

  • Abolfazl Zolfaghari
    • 1
  • Masoud Goharimanesh
    • 2
  • Ali Akbar Akbari
    • 2
    Email author
  1. 1.Department of Mechanical EngineeringTennessee Technological UniversityCookevilleUSA
  2. 2.Department of Mechanical EngineeringFaculty of Engineering, Ferdowsi University of MashhadMashhadIran

Personalised recommendations