Optimal Design of a Heavy Duty Helical Gear Pair Using Particle Swarm Optimization Technique

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Procedia Technology 14 ( 2014 ) 513 – 519

Peer-review under responsibility of the Organizing Committee of ICIAME 2014. doi: 10.1016/j.protcy.2014.08.065

ScienceDirect

2nd International Conference on Innovations in Automation and Mechatronics Engineering,

ICIAME 2014

Optimal Design of a Heavy Duty Helical Gear Pair using Particle

Swarm Optimization Technique

Ketan Tamboli

a

*, Sunny Patel

b

, P.M.George

c

, Rajesh Sanghvi

a

a_{G H Patel College of Engineering & Technology, V V Nagar, Gujarat, India – 388 120} b_{Sr.Engineer, Elecon Engineering Co.Ltd., V U Nagar, Gujarat, India – 388 120}

c_{BVM Engineering Collge.Engineer, V V Nagar, Gujarat, India – 388 120}

Abstract

A helical gear pair of a heavy duty gear reducer is considered for the objective of minimum volume, since the most power transmission systems require low weight energy efficient and cost effective system elements. The various factors for sizing and strength of gears are computed for gear geometry parameters using DIN standard. The formulation of the constrained non-linear multi-variable optimization problem with derived objective function and constraints is presented. The solution is attempted using Particle Swarm Optimization (PSO). The results achieved are satisfactory and helps designer to employ for minimum material and cost by fulfilling the strength and performance requirements.

Selection and/or peer-review under responsibility of the Organizing Committee of ICIAME 2014. Keywords: Helical gear design; Genetic Algorithm; Particle swarm optimization;

1. Introduction

Helical gears are currently being used increasingly in cranes, power plant, sugar mills, automobiles, marine, coal plant, and rolling mills as a power transmitting gear owing to their relatively smooth and silent operation large, load carrying capacity and higher operating speed. Designing highly loaded helical gears for power transmission systems that are good in strength and low level in noise necessitates suitable analytical methods which can easily be put into

* Corresponding author. Tel.: 0919428068059; fax: +9102692236896.

E-mail address: [email protected]

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practice and also give useful information on contact and bending stresses. Many researchers attempted finite element methods to analyse the gears.[1,2,3,4,5]. Chan et al.[6] analysed the gear tooth surface for minimum noise for multiple loads. Padmanabhan et al.[7] used Genetic Algorithm (GA) and Finite Element Analysis (FEA) for gear pair design for optimum space requirements by satisfying bending and crushing stress. The PSO has been used by other researchers [8,9,10,11,12] for optimal design of welded beam, pressure vessel, compression spring, etc. Since the helical gear pair finds ample applications, an attempt is presented here for optimizing the volume of a certain heavy duty application gears.

Nomenclature

mn normal module, mm

b face width, mm

Z1 number of teeth on pinion

Z2 number of teeth on gear

ȕKHOL[DQJOHGHJUHHV

ߝఈ ଵ transverse contact ratio for pinion

ߝఈ ଶ transverse contact ratio for wheel

2. Design Methodology

For a certain application considered in present work based on [13], the design inputs are (a) power to be

transmitted=120 kW (b) gear ratio=5.18 (c) pressure angle=200_{(d) helix angle=12}0_{, (e) material is case hardened}

steel (20MnCr5). By Lewis equation, the module is yielded as 14.

2.1. Calculation of geometrical parameters and strength based factors

Using DIN 3990 and 3960 standards [14,15], various parameters are calculated and shown in Table 2, while in Table 3 the individual parameters for pinion and wheel are shown. Table 4 shows computed strength based factors. All tables are shown in Appendix.

3. Formulation of minimum volume problem

The design variables for the formulation are face width (b or x2), gear teeth (Z1or x3and Z2or x4KHOL[DQJOHȕRU

x5) as mn(or x1) is already computed as 14.

3.1. Formulation of objective function

For the present case of optimization, the objective function is the minimization of volume of cylindrical gear pair given by [13],

2 2 2 1 2 2 4 cos n m b V S Z Z E (1) Representing the objective function in the form as below,

2 2 2 1 2 1 2 3 4 5 2 1 2 , , , , , , , , 4 cos n n m b f x V f m b Z Z E f x x x x x S Z Z E (2)

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3.2. Formulation of constraints

The strength considerations such as (i) factor of safety from pitting (ii) contact stress (iii) contact ratio (iv) number of teeth constraint (v) face width constraint (vi) helix angle constraints etc. form the base for following constraints derivation. The detailed mathematical derivations are not presented due to space constraints. However, the manually calculated parameters are presented in Tables 1, 2 and 3 which are required for the said derivations.

3.2.1 Factor of safety from pitting

The factor of safety constraints of pitting are deduced from contact stress, elasticity, contact ratio and poisson’s ratio for pinion and wheel.

3 4

1 1 5 3 1 1 cos 0.681 tan cos 595.94 6.28 x x x x x D H ª ª ª ººº « « « »»» « « « »»» « « « »»» « « «_¬ »_¼»» ¬ ¼ ¬ ¼₍₃₎

3 4

1 1 5 4 2 1 cos 0.531 tan cos 595.94 6.28 x x x x x D H ª ª ª ººº « « « »»» « « « »»» « « « »»» « «_¬ «_¬ »_¼»_¼» ¬ ¼₍₄₎ ₁

5 1 2 2 1261.82 1.2 0 1 1 189.81 cos 38.251 1.14 g x x x D D H H d u u u u u (5) ₂

5 1 2 2 1262.44 1.2 0 1 1 189.81 cos 38.251 1.14 g x x x D D H H d u u u u u (6)

3.2.2 Factor of safety from tooth breakage

The tooth breakage involves factors like relative toughness, life, size, reliability, application, dynamic load, distribution of load and helix angle for bending stress.

₃

2 5 5 2 1 1 2 1 435.43 1.4 0 sin 280850 0.75 1.818 0.25 1 1.255 120 g x x x x x x H_D H_D Sx d § · § · u u¨ ¸ ¨u u ¸u © ¹ © ¹ (7) ₄

2 5 5 2 1 1 2 1 519.08 1.4 0 sin 280850 0.75 1.818 0.25 1 1.255 120 g x x x x x x H_D H_D Sx d § · § · u u¨ ¸ ¨u u ¸u © ¹ © ¹ (8)

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3.2.3 Number of teeth constraints

The minimum number of tooth constraints are based on avoidance of interference.

g x5

25x3d0 g x6

x356 0d (9)

g x7

130x4d0 g x9

x4290 0d (10)

3.2.4 Face width constraints

The constraints for minimum and maximum face width are based on earlier computations.

g x₉

50x₂d0 g10

x x2250 0d (11)

3.2.5 Helix angle constraints

The helix angle is 4° as minimum [13] to increase load capacity and maximum is based on controlling the axial load and vibrations.

g₁₁

x 4 x₅d0 g12

x x519.5 0d (12)

4. Methods of solution and development of codes

The well-known evolutionary method PSO is used for solving the above formed optimum design problem. The method is outlined in following section.

4.1 Particle Swarm Optimization

Particle Swarm Optimization (PSO) is a population-based Evolutionary Algorithm first introduced by Kennedy and Eberhart [16]. It is stochastic in nature and models itself on flocking behavior such as that of birds. It simulates the social behavior of a group of individuals called particles, in a multi-dimensional search space. Initially a population of particles is randomly generated. These particles are then flown through the problem space by following the current optimum particles. The particles are evolved according to the update equations described by Y.Shi et al. [17] as follows:

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x_id x_idv_id ( 14 )

where ݓ is inertia weight, c and 1 c are learning rates, 2 rand and () Rand are independent random numbers ()

generated uniformly in

_{> @}

0,1 ,p is the _id i particle’s location at which its best fitness has been achieved so far, th pgd

is the location at which the best fitness of the entire population has been achieved so far, x is the current location _id

of the i particle with current velocityth v New velocity id. v is calculated from Equation (13) and hence the new id

position from Equation (14). For simple applications, the suggested values of c and 1 c are 1.49445 and that of 2 wis

0.729. This is called the Gbest version of PSO. C.Worasucheep [11] presented different ways to handle the constraints. In this work, Penalty Function Method is used to handle the constraints.

The computer code is developed in MATLAB. The developed PSO code is experimented with various number of iterations and population size, the results of which are summarized in Table 1. The highlighted values indicate minimum for given number of iterations and population size.

1 2

* * ()* * ()*

id id id id id gd

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Table 1 Results by PSO

Iterations Population size b (mm) Z1 Z2 ȕGHJUHH Volume (×108) (mm3)

500 20 60 48 156 8.3 2.5233 80 38 139 9.9 2.6312 400 20 62 41 138 17.0 2.1621 61 42 142 7.6 2.0970 200 30 64 26 138 8.6 1.9941 71 53 130 14.2 2.3028 100 30 53 30 142 11.9 1.8046 50 43 147 14.2 1.9302 50 30 68 56 141 6.0 2.4443 64 25 151 11.1 2.4163

The illustrative graphs obtained by PSO of volume against (a) face width (b) number of teeth on pinion (c) number of teeth on wheel (d) helix angle are shown in Appendix as Fig.1.

5. Conclusion and way forward

The helical gear pair for heavy duty application is optimized for minimum volume to have light weight, energy efficient and cost effective with improved performance. The application of PSO gives optimum volume as 1.8046

x108 _mm3_{of the pair for which b=53, Z}

1=30, Z2=142 and ȕ=11.9°. The work can be further extended for

multi-objective and use of other evolutionary methods.

Acknowledgements

The encouragement and support provided by Sophisticated Instrumentation Centre for Applied Research and Testing (SICART), an institute sponsored by Department of Science & Technology, Govt. of India, New Delhi jointly by Charutar Vidya Mandal, Vallabh Vidyanagar, Gujarat is highly acknowledged.

Appendix A.

Table 2 Calculated parameters for both pinion and wheel

mn Normal module, mm 14 mt Transverse module, mm 14.313

P Power, kW 120 Į Pressure angle, degrees 200

ȕ Helix angle, degrees 120 _r _Ratio _5.18

Ft Tangential force, N 280850 Fb Permissible bending stress, N/mm2 430

Yv Lewis form factor 0.308 KA Application factor 1.0

ĮȦW Working pressure angle, degrees 20.32 Pt Transverse circular pitch, mm 44.96

Pn Normal circular pitch, mm 43.97 Px Axial pitch, mm 211.543

X Sum of addendum shift co-efficient 0.1093 ȕb Base helix angle, degrees 11.269

Ca Working center distance, mm 710 K Addendum modification factor 1.0

St Transverses tooth thickness 32.89 h Whole depth, mm 33.588

et Space width, mm 12.06 en Normal space width, mm 11.79

ȯĮ Transverses contact ratio 1.578 ȯȕ Axial Contact ratio 1.1587

Fn Normal force, N 298875 Fr Radial force, N 102221

KV Dynamic load factor 1.001 K+Į Transverse load distribution factors 1.0

ZH Zone factor For contact stress 2.429 ZE Elasticity factor for contact stress 189.812

Zȯ Contact ratio factor for contact 0.796 Zȕ Helix angle factor for contact stress 0.989

ZL Life factor for contact stress 1.0 ZI Lubrication factor for contact stress 0.92

Zw Work hardening factor for contact 1.0 Vm Mean velocity, m/sec 0.44

ZR Surface roughness factor for contact 0.9 Įt Transverse pressure angle, degrees 20.41

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Table 3 Separately calculated geometrical parameters for pinion and wheel

Pinion Wheel

Zv Virtual number of teeth 17.007 88.222

d Nominal pitch circle diameter, mm 229 1187

Dw Operating pitch circle diameter, mm 229.495 1190.505

Z Number of teeth 16 83

db Base diameter, mm 214.627 1113.397

dt Tip circle diameter, mm 259.78 1216.19

ha Addendum, mm 15.388 14.118

hd Dedendum, mm 18.200 19.470

Sn Normal tooth thickness, mm 23.01 22.085

Table 4 Strength based factors for pinion and wheel

Pinion Wheel

K+ȕ,K)ȕ Load distribution factors 1.3 1.254

KR Reliability factor 1.015 1.014

Yȯ Contact ratio factor for bending stress 2.74 2.79

YR Factor for relative roughness 0.957 0.957

a b

c d

Fig. 1 Illustrative graphs obtained by PSO of volume against (a) face width (b) number of teeth on pinion (c) number of teeth on wheel (d) helix angle

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Number of teeth on wheel

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