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Optimum design of straight bevel gears pair using evolutionary algorithms

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Journal of the Brazilian Society of Mechanical Sciences and Engineering Aims and scope Submit manuscript

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.

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Abbreviations

A iG :

Gear inner cone distance (mm)

A m :

Mean cone distance (mm)

b :

Face width (mm)

b ilp :

Pinion limit inner dedendum (mm)

b ip :

Pinion inner dedendum (mm)

b p :

Pinion mean dedendum (mm)

b min/b max :

Lower/upper limit range of face width (mm)

d :

Pitch diameter (mm)

e :

Base of natural logarithm

E i /E i−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

K A :

Overload factor

K V :

Dynamic factor

K Hβ :

Load distribution factor

K θ :

Temperature factor

k :

Boltzmann constant

m :

Module (mm)

m et :

Outer transverse module (mm)

n 1 :

Input speed (rpm)

P i :

Acceptance probability of the found solution

P r :

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

Q v :

Transmission accuracy number

r :

Cone pitch (mm)

S H/S F :

Safety factor for contact/bending stress

T i /T (i−1) :

System temperature of the system at current/previous position

V straight bevel gear :

Volume of straight bevel gear (mm3)

Vol:

Total volume of pinion and gear

W t :

Transmitted load (N)

X :

Vector of design variables

Y NT :

Stress-cycle factor for bending strength

Y x :

Reliability factor for bending strength

Y z :

Size factor for bending

Y β :

Length-wise curvature factor for bending strength

z :

Teeth number

z 1/z 2 :

Teeth number of pinion/gear

Z E :

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

Z I :

Contact geometry factor

Z J :

Bending geometry factor

Z NT :

Stress-cycle factor for pitting resistance

Z x :

Size factor for pitting resistance

Z xc :

Crowning factor for pitting

Z w :

Hardness-ratio factor

Z z :

Reliability factor for pitting

α :

Cooling rate of the system

γ :

Pinion pitch angle (rad)

λ :

Cone angle (rad)

σ Hlim/σ Flim :

Allowable contact/bending stress number (N/mm2)

δ 1/δ 2 :

Cone angle of pinion/gear (rad)

δ G/δ P :

Gear/pinion dedendum angle (rad)

Ψ iG :

Inner gear spiral angle (rad)

φ :

Normal pressure angle at pitch surface (rad)

φ Ti :

Inner transverse pressure angle (rad)

AGMA:

American Gear Manufacturers Association

GA:

Genetic algorithm

min:

Minimum

N:

No

SA:

Simulated annealing

TA:

Traditional algorithm

Y:

Yes

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Authors and Affiliations

  1. Department of Mechanical Engineering, Tennessee Technological University, 1 William L Jones Dr, TN 38505, Cookeville, USA

    Abolfazl Zolfaghari

  2. Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

    Masoud Goharimanesh & Ali Akbar Akbari

Authors
  1. Abolfazl Zolfaghari
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  2. Masoud Goharimanesh
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  3. Ali Akbar Akbari
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Corresponding author

Correspondence to Ali Akbar Akbari.

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Technical Editor: Fernando Antonio Forcellini.

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Zolfaghari, A., Goharimanesh, M. & Akbari, A.A. Optimum design of straight bevel gears pair using evolutionary algorithms. J Braz. Soc. Mech. Sci. Eng. 39, 2121–2129 (2017). https://doi.org/10.1007/s40430-017-0733-9

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  • DOI: https://doi.org/10.1007/s40430-017-0733-9

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