Modelling and Analysis of the Cone Coupling Problem Using Optimization

Modelling and Analysis of the Cone Coupling Problem Using Optimization

Mubina Nancy Elizabeth Amudhini Stephen

Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore 641114, India

Corresponding Author Email: 
elizi.felix@gmail.com
Page: 
1385-1392
|
DOI: 
https://doi.org/10.18280/mmep.090529
Received: 
20 April 2022
|
Revised: 
9 June 2022
|
Accepted: 
17 June 2022
|
Available online: 
13 December 2022
| Citation

© 2022 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).

OPEN ACCESS

Abstract: 

A coupling is a mechanism that transmits operative power between two shafts that are revolving at different speeds. A coupling connects two shafts at their ends and can slip or fail depending on the torque limit. It is an essential component of any power transmission system and may survive for a very long time if properly designed and maintained. This study's current research, a newly developed optimization algorithms are used to minimize the volume of cone coupling. The current study presented here compares modern metaheuristic methods for optimizing the design of the cone coupling problem. The algorithms used are particle swarm optimization (PSO), crow search algorithm (CSA), enhanced honeybee mating optimization (EHBMO), Harmony search algorithm (HSA), Krill heard algorithm (KHA), Pattern search algorithm (PSA), Charged system search algorithm (CSSA), Salp swarm algorithm (SSA), Big bang big crunch optimization (B-BBBCO), Gradient based Algorithm (GBA). The performance of these algorithms is assessed both statistically and subjectively. The algorithms' performance is evaluated quantitatively and qualitatively using consistency, simplicity and quality. The experimental results on the cone clutch problem shows that PSO produces greater results than EHBMO, whereas CSA and BBCO produce approximately identical results.

Keywords: 

cone coupling, optimization analysis, mathematical modelling, torque, axial force, flanges

1. Introduction

Cone coupling are a form of friction clutch that engages and disengages the engine shaft from the transmission box shaft when the gear ratio changes. It is one of the oldest clutches in use in the vehicle industry. In comparison to positive displacement clutches, which were utilised before to the discovery of friction clutches, this clutch is simple to engage and disengage. Because of the larger contact area, a cone clutch can transfer more torque than a plate clutch of the same size. When a large amount of torque needs to be transferred at a low rotational speed, this clutch is used. The goal of the optimization problem is to reduce the volume design of the cone clutch such that it can transmit a specified minimum torque. Couplings are used to connect two pieces of rotating equipment while allowing for some degree of misalignment, end movement, or both. Cone coupling is shown in the Figure 1.

Figure 1. Cone coupling

A coupling can also be a mechanical mechanism that connects the ends of nearby pieces or objects in a broader sense. Couplings generally do not allow shafts to be disconnected during operation, however torque-limiting couplings can slip or disconnect if a torque limit is exceeded. Couplings can be chosen, installed, and maintained in such a way that maintenance time and expense are decreased.

2. Literature Survey

Jovanović et al. [1] implements the applications of grasshopper optimization in mechanical engineering. He shows how the Grasshopper Optimization Algorithm (GOA) can be utilised to solve specific engineering optimization problems. Li et al. [2] introduced the light adaption, The sensitivity and maximum amplitude (Rmax) of the mouse photopic electroretinogram (ERG) b-wave alter as a result of light adaptation. We investigated how manipulation of gap junctional coupling between rod and cones affects the light-adapted ERG using the ERG. Muchungi and Casey [3] is the first to suggest a cone simulator that incorporates rod input. Alizadeh et al. [4] tackles the issue of managing the lubrication regime in sliding lubricated surfaces in order to reduce wear and extend the lifespan of the friction lining material. Vyrabov [5] introduce the cone clutches and friction drives with wedge-type bodies have a circumferential force limit. Milenković [6] offers the fundamentals of a metaheuristic algorithm based on the behaviour of Harris hawks are demonstrated. Milenković and Jovanović [7] also shows how the Grasshopper Optimization Algorithm (GOA) can be utilised to solve specific engineering optimization problems. Particle Swarm Optimization was used to do parametric optimization on the spring design problem, pressure vessel design problem, cantilever beam design problem, cone coupling design problem, and welded beam design problem is also introduced by Milenković et al. [8]. Gordy demonstrated SMAP (Soil Moisture Active and Passive) Cone Clutch Assembly (CCA) Thermal Conductance Test [9]. Nguyen et al. [10] introduced for the bias ratio and noise condition, the surface topology, cone angle, and forces acting on the cone of the clutch type limited slip differential (LSD) are important design characteristics. Chase [11] introduces d the goals of this paper are to collect information about clutch design in the United States and the United Kingdom, compare the advantages and disadvantages of various clutch types, and provide some notes on clutch theory without attempting a comprehensive treatment of the numerous factors involved. Genetic algorithm is used for optimization by Fadah et al. [12]. Volume of the fin shape is optimized by Nguyen et al. [10]. Therefore this cone coupling design problem is done by many researchers. In this study, cone coupling is optimized by different types of optimization algorithms.

3. Mathematical Modelling

The cone coupling's design. The goal of this optimization problem is to reduce the coupling volume to allow momentum transfer. The inner radius of the connection R1 and the outer radius of the coupling R2 are the problem variables [13].

The minimum volume design of the cone coupling can transmit a specified torque.

By selecting the outer and inner radii of the cone R1 and R2, as design variable, the objective function can be expressed as:

$f\left(R_1, R_2\right)=\frac{1}{3} \pi h\left(R_1^2+R_1 R_2+R_2^2\right)$          (1)

where, the axial thickness, h is given by

$h=\frac{R_1-R_2}{\tan \alpha}$          (2)

Eqns. (1) and (2) yield:

$f\left(R_1, R_2\right)=k_1\left(R_1^3-R_2^3\right)$         (3)

where,

$k_1=\frac{\pi}{3 \tan \alpha}$          (4)

k1 is expressed in Eqns. (3) and (4). The axial force applied (F) and the torque developed (T) are given:

$F=\int p d A \sin \alpha=\int_{R_2}^{R_1} p \frac{2 \pi r d r}{\sin \alpha} \sin \alpha=\pi p\left(R_1^2-R_2^2\right)$          (5)

$F=\int r f p d A=\int_{R_2}^{R_1} r f p \frac{2 \pi r}{\sin \alpha} d r=\frac{2 \pi r}{3 \sin \alpha}\left(R_1^3-R_2^3\right)$          (6)

where, p is the pressure, f the coefficient of friction, and A the area of contact. Substitution of p from Eq. (5) into (6) leads to Eqns. (7) and (8):

$T=\frac{k_2\left(R_1^2+R_1 R_2+R_2^2\right)}{R_1+R_2}$          (7)

where,

$k_2=\frac{2 F f}{3 \sin \alpha}$          (8)

Since k1 is a constant, the objective function can be taken as $f=\left(R_1^3-R_2^3\right)$ in Eq. (9). The minimum torque to be transmitted is assumed to be 5k2. In addition, the outer radius R1 is assumed to be equal to at least twice the inner radius R2 [14].

Thus, the optimization problems become objective function

$f\left(R_1, R_2\right)=\left(R_1^3-R_2^3\right)$          (9)

Subject to

$g_1\left(R_1, R_2\right)=\frac{R_1}{R_2} \geq 2$          (10)

$g_2\left(R_1, R_2\right)=\frac{\left(R_1^2+R_1 R_2+R_2^2\right)}{\left(R_1+R_2\right)} \geq 5$          (11)

$1 \leq R_1, R_2 \leq 10$          (12)

Eqns. (10) and (11) and (12) expresses the constraints.

4. Optimization Algorithm

The following are some of the most common problems with traditional gradient methods and direct approaches:

  • It converges to an ideal solution based on the original solution; most algorithms have a propensity to limit themselves to the sub-optimal option.
  • An algorithm that solves one problem may not be efficient when applied to another.
  • When dealing with problems involving nonlinear objectives, discrete variables, and a large number of restrictions, algorithms are inefficient.
  • On a parallel computer, algorithms cannot be employed efficiently.

Addressing large-scale difficulties with nonlinear objectives functions is difficult using standard techniques like steepest descent, dynamic programming, and linear programming. Traditional algorithms can't address non-differentiable problems since they rely on gradient information. In some optimization problems, there are a lot of local optima. As a result of this issue, more powerful optimization approaches are required, and our non-traditional optimization method has been discovered through research.

The following non-traditional optimization algorithms are used which is shown in the Table 1.

(1) Particle swarm optimization (PSO); (2) Crow search algorithm (CSA); (3) Enhanced honeybee mating optimization (EHBMO); (4) Harmony search algorithm (HSA); (5) Krill heard algorithm (KHA); (6) Pattern search algorithm (PSA); (7) Charged system search algorithm (CSSA); (8) Salp swarm algorithm (SSA); (9) Big bang big crunch optimization (B-BBBCO); (10) Gradient based Algorithm (GBA).

Table 1. Optimization algorithms

Optimization Algorithms

Methods

Swarm Intelligence Algorithm

(1) Particle Swarm Optimization (PSO); (2) Crow Search Algorithm (CSA)

(3) Enhanced Honeybee Mating Optimization (EHBMO); (4) Krill Heard Algorithm (KHA)

(5) Salp Swarm Algorithm (SSA)

Physical Related Algorithm

(1) Harmony Search Algorithm (HSA); (2) Charged System Search Algorithm (CSSA)

(3) Big Bang Big Crunch Optimization (B-BBBCO)

Mathematical Programming

Gradient Based Algorithm (GBA)

String Searching Algorithm

Pattern Search Algorithm

5. Methodology

The non-traditional algorithm's performance will vary with each run, but the solution will always be global optimal [15]. As a result, twenty trail runs in all algorithms were performed for each problem, and the average value of the answer was calculated from all the trails. Table 2 shows the specific parameters for several techniques, whereas Table 3 shows the Functional Evaluation FEs number and Number of population NP size.

Table 2. Specific parameter settings of used algorithms

Algorithm

Parameter Settings

PSO

wmin=0.9, wmax=0.4, c1=2, c2=2

CSA

c1=c2=c3=2, ω=0.5, AP=0.2, fl=2, Vmax=[2]D

EHBMO

No. of drones=40, No. of broods=10, No. of selected genes in crossover=8

HAS

HMS=50, HMCR=0.5 fixed, PAR=0.5

KHA

Nmax=0.01, Vf=0.02, Dmax=0.005

PSA

Only the common parameters (Fes and NP)

CSSA

rand-Random value between [0,1], c=0.1, ɛ=0.001

SSA

Only the common parameters (Fes and NP)

B-BBCO

Npop=100, kls=30, α=0.8, Ns=5

GBA

Only the common parameters (Fes and NP)

wmin, wmax are respectively the min and max inertia weight 0.4, c1and c2 are acceleration factors. HMS-Harmony Memory Size, PAR-Pitch Adjustment rate, HMCR-Harmony Memory Consideration rate, Npop- Population size, Kls- no. of non-improvement iteration, α- Reduction rate, Ns -no. of neighbours created in each generation, Nmax- Maximum induced speed, Vf- The foraging speed, Dmax- The maximum diffusion speed, c1, c2, c3- acceleration, ω- inertia weight, fl- length of the crow’s flight, AP- perceptual probability of crow, Vmax- upper limit of the particle update velocity.

Table 3. FEs number and the NP size for the algorithms

Problem

NP

tmax

Fes

Cone coupling

20

250

5000

Tables 3-6 and Figures 2-6 display the values of the outer radius (R1), the inner radius (R2), and the two constraints (g1) and (g2). Table 7 and Figure 7 shows the values of volume minimization.

5.1 The inner radius R1

The inner radius if the cone coupling is minimized by optimizing for 20 trails with 10 different optimization methods.

5.2 The outer radius R2

The outer radius if the cone coupling is minimized by optimizing for 20 trails with 10 different optimization methods.

5.3 Constraint g1

The outer radius is assumed to be equal to at least twice the inner radius.

5.4 Constraint g2

The torque of the shaft is minimized with 20 trails by using different types of optimization methods.

5.5 Volume minimization

Volume of the cone coupling is minimized by the optimization methods for 20 trails.

In Table 8, a comparison of results for design of cone coupling optimization problem are shown. Analysing the table results a conclusion has been drawn that the PSO gives better results in comparison to EHBMO, while in comparison to CSA and BBCO the results are nearly the same.

Figure 2. The inner radius R1

Figure 3. The outer radius R2

Figure 4. Constraint g1

Figure 5. Constraint g2

Figure 6. Volume Minimization fmin

Figure 7. The inner radius of the Coupling R1

Figure 8. Constraint g1

Figure 9. The outer radius of the Coupling R2

Figure 10. Constraint g2

Figure 11. Volume minimization

Table 4. The inner radius R1

Trial

PSO

EHDMO

HSA

CSA

CSSA

BBCO

GBA

KHA

PSA

SSA

1

4.2785

4.3256

4.3089

4.2871

4.2859

4.2984

4.3059

4.2985

4.2998

4.2987

2

4.2785

4.3265

4.3025

4.2871

4.2956

4.2984

4.3152

4.2981

4.2998

4.2989

3

4.2785

4.3256

4.3021

4.2871

4.2956

4.2984

4.3058

4.2965

4.2998

4.299

4

4.2785

4.3256

4.3025

4.2871

4.2969

4.2984

4.3056

4.2958

4.2998

4.2991

5

4.2785

4.3256

4.3025

4.2871

4.2989

4.2984

4.3059

4.2987

4.2998

4.2987

6

4.2785

4.3256

4.3056

4.2871

4.2965

4.2984

4.3059

4.2931

4.2998

4.2986

7

4.2785

4.3256

4.3052

4.2871

4.2986

4.2984

4.3059

4.2951

4.2998

4.2989

8

4.2785

4.3265

4.3089

4.2871

4.2963

4.2984

4.3059

4.2965

4.2998

4.2987

9

4.2785

4.3215

4.3052

4.2871

4.2958

4.2984

4.3058

4.2988

4.2998

4.2986

10

4.2785

4.3215

4.3021

4.2871

4.2965

4.2984

4.3058

4.2899

4.2998

4.2985

11

4.2785

4.3256

4.3025

4.2871

4.2936

4.2984

4.3058

4.2985

4.2998

4.2982

12

4.2785

4.3256

4.3025

4.2871

4.2965

4.2984

4.3058

4.2889

4.2998

4.2983

13

4.2785

4.3265

4.3025

4.2871

4.2956

4.2984

4.3059

4.2965

4.2998

4.2985

14

4.2785

4.3256

4.3025

4.2871

4.2986

4.2984

4.3059

4.2954

4.2998

4.2987

15

4.2785

4.3214

4.3025

4.2871

4.2989

4.2984

4.3059

4.2963

4.2998

4.2986

16

4.2785

4.3256

4.3021

4.2871

4.2969

4.2984

4.3058

4.2951

4.2998

4.2987

17

4.2785

4.3215

4.3021

4.2871

4.2965

4.2984

4.30587

4.2971

4.2998

4.2981

18

4.2785

4.3256

4.3025

4.2871

4.2965

4.2984

4.3059

4.2985

4.2998

4.298

19

4.2785

4.3266

4.3021

4.2871

4.2968

4.2984

4.3059

4.2985

4.2998

4.2888

20

4.2785

4.3256

4.3025

4.2871

4.2965

4.2984

4.3059

4.2951

4.2998

4.2889

Average

4.2785

4.3250

4.3035

4.2871

4.2962

4.2984

4.3063

4.2960

4.2998

4.2976

Max

4.2785

4.3266

4.3089

4.2871

4.2989

4.2984

4.3152

4.2988

4.2998

4.2991

Min

4.2785

4.3214

4.3021

4.2871

4.2859

4.2984

4.3056

4.2889

4.2998

4.2888

SD

0.0000

0.0018

0.0021

0

0.0027

0

0.0020

0.0027

0

0.0030

Fes

5000

5000

5000

5000

5000

5000

5000

5000

5000

5000

Table 5. The outer radius R2

Trial

PSO

EHDMO

HSA

CSA

CSSA

BBCO

GBA

KHA

PSA

SSA

1

2.1327

2.13426

2.1457

2.1344

2.1429

2.1468

2.1429

2.143

2.1489

2.1405

2

2.1327

2.1343

2.1424

2.1344

2.14299

2.1468

2.1427

2.1452

2.1489

2.1409

3

2.1327

2.13426

2.1456

2.1344

2.14563

2.1468

2.14285

2.1456

2.1489

2.1409

4

2.1327

2.1343

2.1457

2.1344

2.14296

2.1468

2.14283

2.1466

2.1489

2.1406

5

2.1327

2.13426

2.1452

2.1344

2.1457

2.1468

2.14285

2.1465

2.1489

2.1411

6

2.1327

2.1342

2.1497

2.1344

2.14563

2.1468

2.14287

2.142

2.1489

2.1415

7

2.1327

2.1343

2.1499

2.1344

2.14296

2.1468

2.1429

2.1452

2.1489

2.1416

8

2.1327

2.13426

2.149

2.1344

2.1457

2.1468

2.1429

2.14

2.1489

2.1402

9

2.1327

2.13429

2.1459

2.1344

2.1457

2.1468

2.1429

2.1456

2.1489

2.1452

10

2.1327

2.13425

2.1456

2.1344

2.1457

2.1468

2.1429

2.14

2.1489

2.1456

11

2.1327

2.13422

2.1456

2.1344

2.14563

2.1468

2.14292

2.143

2.1489

2.1458

12

2.1327

2.13425

2.149

2.1344

2.14524

2.1468

2.1459

2.1423

2.1489

2.1453

13

2.1327

2.13426

2.149

2.1344

2.1459

2.1468

2.14265

2.1432

2.1489

2.1442

14

2.1327

2.13426

2.1486

2.1344

2.1452

2.1468

2.14266

2.142

2.1489

2.1443

15

2.1327

2.13426

2.149

2.1344

2.1452

2.1468

2.14562

2.1432

2.1489

2.1436

16

2.1327

2.13426

2.1459

2.1344

2.14256

2.1468

2.14524

2.1403

2.1489

2.1463

17

2.1327

2.13425

2.1457

2.1344

2.12546

2.1468

2.14587

2.1431

2.1489

2.1456

18

2.1327

2.13425

2.1457

2.1344

2.1457

2.1468

2.14567

2.1426

2.1489

2.1432

19

2.1327

2.13426

2.1467

2.1344

2.1457

2.1468

2.14287

2.1452

2.1489

2.1452

20

2.1327

2.13426

2.149

2.1344

2.1457

2.1468

2.14567

2.1456

2.1489

2.1455

Average

2.1327

2.13426

2.1469

2.1344

2.14391

2.1468

2.14368

2.14351

2.1489

2.143355

Max

2.1327

2.1343

2.1499

2.1344

2.1459

2.1468

2.1459

2.1466

2.1489

2.1463

Min

2.1327

2.1342

2.1424

2.1344

2.12546

2.1468

2.14265

2.14

2.1489

2.1402

SD

0

2.3E-05

0.002

0

0.00451

0

0.00134

0.00209

0

0.002188

Fes

5000

5000

5000

5000

5000

5000

5000

5000

5000

5000

Table 6. Constraint g1

Trial

PSO

EHDMO

HSA

CSA

CSSA

BBCO

GBA

KHA

PSA

SSA

1

2.006

2.0268

2.0082

2.0086

2

2.00224

2.0094

2.0058

2.00093

2.00827

2

2.006

2.0271

2.0083

2.0086

2.0045

2.00224

2.0139

2.0036

2.00093

2.00799

3

2.006

2.0268

2.0051

2.0086

2.002

2.00224

2.0094

2.0025

2.00093

2.00803

4

2.006

2.0267

2.0052

2.0086

2.0052

2.00224

2.0093

2.0012

2.00093

2.00836

5

2.006

2.0268

2.0056

2.0086

2.0035

2.00224

2.0095

2.0027

2.00093

2.00771

6

2.006

2.0268

2.0029

2.0086

2.0025

2.00224

2.0094

2.0042

2.00093

2.00728

7

2.006

2.0267

2.0026

2.0086

2.0059

2.00224

2.0094

2.0022

2.00093

2.00733

8

2.006

2.0272

2.0051

2.0086

2.0023

2.00224

2.0094

2.0077

2.00093

2.00855

9

2.006

2.0248

2.0063

2.0086

2.0021

2.00224

2.0094

2.0035

2.00093

2.00382

10

2.006

2.0249

2.0051

2.0086

2.0024

2.00224

2.0094

2.0046

2.00093

2.0034

11

2.006

2.0268

2.0053

2.0086

2.0011

2.00224

2.0093

2.0058

2.00093

2.00308

12

2.006

2.0268

2.0022

2.0086

2.0028

2.00224

2.0066

2.002

2.00093

2.00359

13

2.006

2.0272

2.0021

2.0086

2.0018

2.00224

2.0096

2.0047

2.00093

2.00471

14

2.006

2.0268

2.0025

2.0086

2.0038

2.00224

2.0096

2.0053

2.00093

2.00471

15

2.006

2.0248

2.0022

2.0086

2.004

2.00224

2.0068

2.0046

2.00093

2.00532

16

2.006

2.0268

2.0048

2.0086

2.0055

2.00224

2.0072

2.0068

2.00093

2.00284

17

2.006

2.0249

2.005

2.0086

2.0215

2.00224

2.0066

2.0051

2.00093

2.00322

18

2.006

2.0268

2.0052

2.0086

2.0024

2.00224

2.0068

2.0062

2.00093

2.00541

19

2.006

2.0272

2.0041

2.0086

2.0026

2.00224

2.0095

2.0038

2.00093

1.99925

20

2.006

2.0268

2.0022

2.0086

2.0024

2.00224

2.0068

2.0018

2.00093

1.99902

Average

2.006

2.0265

2.0045

2.0086

2.0039

2.00224

2.0089

2.0042

2.00093

2.00509

Max

2.006

2.0272

2.0083

2.0086

2.0215

2.00224

2.0139

2.0077

2.00093

2.00855

Min

2.006

2.0248

2.0021

2.0086

2

2.00224

2.0066

2.0012

2.00093

1.99902

SD

0

0.0009

0.0019

0

0.0044

0

0.0017

0.0018

0

0.00288

Fes

5000

5000

5000

5000

5000

5000

5000

5000

5000

5000

Table 7. Constraint g2

Trial

PSO

EHDMO

HSA

CSA

CSSA

BBCO

GBA

KHA

PSA

SSA

1

5.011

5.056

5.0413

5.0195

5.0192

5.03149

5.0382

5.0312

5.03303

5.03112

2

5.011

5.0569

5.035

5.0195

5.0285

5.03149

5.0471

5.031

5.03303

5.03135

3

5.011

5.056

5.0349

5.0195

5.0288

5.03149

5.0382

5.0296

5.03303

5.03144

4

5.011

5.0561

5.0353

5.0195

5.0297

5.03149

5.0379

5.029

5.03303

5.03151

5

5.011

5.056

5.0353

5.0195

5.0319

5.03149

5.0382

5.0317

5.03303

5.03118

6

5.011

5.056

5.0386

5.0195

5.0297

5.03149

5.0382

5.026

5.03303

5.03113

7

5.011

5.056

5.0383

5.0195

5.0313

5.03149

5.0383

5.0282

5.03303

5.03142

8

5.011

5.0569

5.0417

5.0195

5.0294

5.03149

5.0382

5.029

5.03303

5.03109

9

5.011

5.0521

5.0379

5.0195

5.029

5.03149

5.0382

5.0317

5.03303

5.03151

10

5.011

5.0521

5.0349

5.0195

5.0297

5.03149

5.0382

5.0227

5.03303

5.03146

11

5.011

5.056

5.0353

5.0195

5.0269

5.03149

5.0382

5.0312

5.03303

5.03119

12

5.011

5.0561

5.0357

5.0195

5.0296

5.03149

5.0385

5.022

5.03303

5.03124

13

5.011

5.0569

5.0356

5.0195

5.0288

5.03149

5.0382

5.0293

5.03303

5.03131

14

5.011

5.0561

5.0356

5.0195

5.0316

5.03149

5.0382

5.0281

5.03303

5.03151

15

5.011

5.0521

5.0357

5.0195

5.0319

5.03149

5.0385

5.0291

5.03303

5.03134

16

5.011

5.056

5.0349

5.0195

5.0297

5.03149

5.0384

5.0277

5.03303

5.03172

17

5.011

5.0521

5.0349

5.0195

5.0275

5.03149

5.0385

5.0299

5.03303

5.03108

18

5.011

5.056

5.0353

5.0195

5.0297

5.03149

5.0385

5.0311

5.03303

5.03073

19

5.011

5.057

5.035

5.0195

5.0299

5.03149

5.0383

5.0314

5.03303

5.02222

20

5.011

5.0561

5.0357

5.0195

5.0297

5.03149

5.0385

5.0282

5.03303

5.02234

Average

5.011

5.0554

5.0364

5.0195

5.0291

5.03149

5.0387

5.0289

5.03303

5.03039

Max

5.011

5.057

5.0417

5.0195

5.0319

5.03149

5.0471

5.0317

5.03303

5.03172

Min

5.011

5.0521

5.0349

5.0195

5.0192

5.03149

5.0379

5.022

5.03303

5.02222

SD

0

0.0017

0.0021

0

0.0027

0

0.002

0.0027

0

0.00278

Fes

5000

5000

5000

5000

5000

5000

5000

5000

5000

5000

Table 8. Volume minimization

Trial

PSO

EHDMO

HSA

CSA

CSSA

BBCO

GBA

KHA

PSA

SSA

1

68.5698

71.2157

70.123

69.06997

68.8872

69.52422

69.9945

69.5822

69.57278

69.6277

2

68.5698

71.2651

69.817

69.06997

69.423

69.52422

70.5194

69.5297

69.57278

69.63328

3

68.5698

71.2157

69.749

69.06997

69.3883

69.52422

69.9934

69.4355

69.57278

69.63883

4

68.5698

71.217

69.771

69.06997

69.4989

69.52422

69.98

69.383

69.57278

69.6485

5

68.5698

71.2156

69.777

69.06997

69.5725

69.52422

69.9986

69.545

69.57278

69.61945

6

68.5698

71.2164

69.886

69.06997

69.4405

69.52422

69.9962

69.297

69.57278

69.6084

7

68.5698

71.2151

69.861

69.06997

69.5914

69.52422

69.9993

69.3635

69.57278

69.62366

8

68.5698

71.2699

70.081

69.06997

69.426

69.52422

69.9958

69.5127

69.57278

69.63182

9

68.5698

70.9861

69.918

69.06997

69.4008

69.52422

69.9924

69.563

69.57278

69.55741

10

68.5698

70.9866

69.748

69.06997

69.439

69.52422

69.9924

69.1477

69.57278

69.54634

11

68.5698

71.2168

69.771

69.06997

69.2775

69.52422

69.9925

69.5822

69.57278

69.52695

12

68.5698

71.2194

69.725

69.06997

69.4424

69.52422

69.9525

69.0609

69.57278

69.5394

13

68.5698

71.2667

69.724

69.06997

69.3871

69.52422

69.9979

69.4686

69.57278

69.56566

14

68.5698

71.2194

69.731

69.06997

69.5604

69.52422

69.9978

69.4243

69.57278

69.57537

15

68.5698

70.9812

69.725

69.06997

69.5777

69.52422

69.958

69.4576

69.57278

69.57948

16

68.5698

71.2157

69.745

69.06997

69.5043

69.52422

69.9606

69.431

69.57278

69.54776

17

68.5698

70.987

69.748

69.06997

69.7128

69.52422

69.9519

69.5032

69.57278

69.52417

18

68.5698

71.2158

69.77

69.06997

69.4391

69.52422

69.9563

69.5877

69.57278

69.55174

19

68.5698

71.2731

69.734

69.06997

69.4563

69.52422

70.0002

69.5519

69.57278

69.01539

20

68.5698

71.2176

69.725

69.06997

69.4396

69.52422

69.9563

69.358

69.57278

69.01677

Average

68.5698

71.1808

69.806

69.06997

69.4432

69.52422

70.0093

69.4392

69.57278

69.5289

Max

68.5698

71.2731

70.123

69.06997

69.7128

69.52422

70.5194

69.5877

69.57278

69.6485

Min

68.5698

70.9812

69.724

69.06997

68.8872

69.52422

69.9519

69.0609

69.57278

69.01539

SD

0

0.10246

0.1156

0

0.1615

0

0.12149

0.14237

0

0.179956

Fes

5000

5000

5000

5000

5000

5000

5000

5000

5000

5000

Table 9. Comparison of the best optimum solution for the cone coupling problem

Variables

PSO

EHBMO

HAS

CSA

CSSA

BBCO

GBA

KHA

PSA

SSA

R1 m

4.2786

4.32501

4.3035

4.2871

4.29622

4.2984

4.30636

4.29605

4.2998

4.29763

R2 m

2.1327

2.13426

2.1469

2.1344

2.14391

2.1468

2.14368

2.14351

2.1489

2.143355

g1 m

2.0062

2.02647

2.0045

2.00857

2.00393

2.00224

2.00886

2.00421

2.00093

2.00509

g2 m

5.0112

5.05543

5.0364

5.01948

5.02912

5.03149

5.03871

5.02891

5.03303

5.03039

Volume

68.57

71.1808

69.807

69.07

69.4432

69.5242

70.0093

69.4392

69.5728

69.5289

Table 10. Statistical result of the used algorithms for the cone coupling problem

Algorithm

Best

Mean

Worst

SD

Fes

PSO

68.57

68.57

68.57

0

5000

EHDMO

70.0071

70.2176

70.336

0.11403

5000

HSA

69.7241

69.8065

70.123

0.11559

5000

CSA

69.07

69.07

69.07

0

5000

CSSA

68.8872

69.4432

69.713

0.1615

5000

BBCO

69.5242

69.5242

69.524

0

5000

GBA

69.9519

70.0093

70.519

0.12149

5000

KHA

69.0609

69.4392

69.588

0.14237

5000

PSA

69.5728

69.5728

69.573

0

5000

SSA

69.0154

69.5289

69.649

0.17996

5000

6. Result and Discussion

6.1 Consistency

The consistency table gives the parameters that remain constant for all the trails. All the solvers give the value of PSO, CSA, BBCO and HSA for all the runs, which in turn indicates that the requirements are in the acceptable range.

  • R1 - PSO (4.2785698567), CSA (4.2871), BBCO (4.2984)
  • R2 - PSO (2.1326522330), CSA (2.13440), HSA (2.1490)
  • g1 - PSO (2.0062201377), EHBMO (1.6366), HSA (2.00450)
  • g2 - PSO (5.0112134936), HSA (5.036353), CSA (5.01948)

So, we see that the solvers PSO, CSA, BBCO, PSA remains constant throughout their runs.

6.2 Simplicity of algorithm

Of all the algorithm, we have taken PSO is the simplest followed by EHBMO, SSA, HSA, BBCO.

6.3 Minimum values of variables

The best optimal solution and statistical simulation results for the cone coupling problem are presented in Table 9, Table 10, and Figures 7-11. Table 8 shows that all of the methodologies used are capable of finding a globally feasible solution. However, with standard deviation values of 0, the PSO algorithm is the most robust in handling this problem, followed by EHBMO, SSA, HAS, CSA, CSSA, BBCO, PSA, GBA, and KHA.

  • R1 - PSO (4.2786), CSA (4.2871), BBCO (4.2984)
  • R2 - PSO (2.178655), CSA (2.13440), HSA (2.1490)
  • g1 - PSO (1.962768), EHBMO (1.6366), HSA (2.00450)
  • g2 - PSO (5.016092), HSA (5.036353), CSA (5.01948)
7. Conclusions

In this study, the volume of the cone coupling is optimized, and these optimized results are validated using ANSYS simulation. This volume minimized cone coupling will be very reachable for small scale industries and it gives profit and gives more manufactures. The following are some of the most common problems with classic gradient methods and traditional direct approaches:

  • It converges to an optimal solution based on the original solution chosen.
  • Most algorithms are prone to limiting themselves to a sun-optimal answer.
  • A problem solved by one algorithm may not be efficient when applied to another.
  • Algorithms are inefficient for solving problems with non-linear objectives, discrete variables, and a large number of restrictions.
  • On a parallel computer, algorithms cannot be employed efficiently.

In general, standard techniques such as steepest descent, dynamic programming, and linear programming make it difficult to address large-scale issues with nonlinear objectives functions. Traditional algorithms cannot address non-differentiable problems because they require gradient information. Some optimization problems have a large number of local optima. As a result of this issue, there is a need to build more powerful optimization approaches, and research has discovered our non-traditional optimization [16, 17].

In this paper, we compared 10 meta-heuristic algorithms to solve the cone coupling. The algorithms used are particle swarm optimization (PSO), crow search algorithm (CSA), enhanced honeybee mating optimization (EHBMO), Harmony search algorithm (HSA), Krill heard algorithm (KHA), Pattern search algorithm (PSA), Charged system search algorithm (CSSA), Salp swarm algorithm (SSA), Big bang big crunch optimization (B-BBBCO), Gradient based Algorithm (GBA). These algorithm’s performance is evaluated statistically and subjectively.

By comparing these methods, we’ve proved that PSO is the best optimization method comparing with other nine methods which we discussed in the result analysis. To minimize the volume of the cone coupling, Particle Swarm Optimization (PSO) got the minimum value comparing with Enhanced Honey-Bee Mating (EHBMO) and Salp Swarm Optimization (SSA). Therefore, for cone coupling problem, Particle Swarm Optimization (PSO) is the best method. These results will be validated using simulation by ANSYS.

The original volume is reduced and this can be sent to the industries. So that will be very reachable for small scale industries and it gives profit and gives more manufactures.

Acknowledgment

The author would like to thank the Karunya Institute of Technology and Sciences for their assistance in carrying out this research.

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