© 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/).
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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.
cone coupling, optimization analysis, mathematical modelling, torque, axial force, flanges
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.
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 (R_{max}) 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.
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 R_{1} and the outer radius of the coupling R_{2} 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 R_{1} and R_{2}, 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)
k_{1} 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 k_{1} 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 5k_{2}. In addition, the outer radius R_{1 }is assumed to be equal to at least twice the inner radius R_{2} [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.
The following are some of the most common problems with traditional gradient methods and direct approaches:
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 |
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 |
w_{min}=0.9, w_{max}=0.4, c_{1}=2, c_{2}=2 |
CSA |
c_{1}=c_{2}=c_{3}=2, ω=0.5, AP=0.2, fl=2, V_{max}=[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 |
N^{max}=0.01, V_{f}=0.02, D^{max}=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 |
N_{pop}=100, k_{ls}=30, α=0.8, N_{s}=5 |
GBA |
Only the common parameters (Fes and NP) |
w_{min}, w_{max} are respectively the min and max inertia weight 0.4, c_{1}and c_{2} are acceleration factors. HMS-Harmony Memory Size, PAR-Pitch Adjustment rate, HMCR-Harmony Memory Consideration rate, N_{pop}- Population size, K_{ls}- no. of non-improvement iteration, α- Reduction rate, N_{s }-no. of neighbours created in each generation, N^{max}- Maximum induced speed, V_{f}- The foraging speed, D^{max}- The maximum diffusion speed, c_{1}, c_{2}, c_{3}- acceleration, ω- inertia weight, fl- length of the crow’s flight, AP- perceptual probability of crow, V_{max}- upper limit of the particle update velocity.
Table 3. FEs number and the NP size for the algorithms
Problem |
NP |
t_{max} |
Fes |
Cone coupling |
20 |
250 |
5000 |
Tables 3-6 and Figures 2-6 display the values of the outer radius (R_{1}), the inner radius (R_{2}), and the two constraints (g_{1}) and (g_{2}). Table 7 and Figure 7 shows the values of volume minimization.
5.1 The inner radius R_{1}
The inner radius if the cone coupling is minimized by optimizing for 20 trails with 10 different optimization methods.
5.2 The outer radius R_{2}
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 R_{1}
Figure 3. The outer radius R_{2}
Figure 4. Constraint g_{1}
Figure 5. Constraint g2
Figure 6. Volume Minimization fmin
Figure 7. The inner radius of the Coupling R_{1}
Figure 8. Constraint g_{1}
Figure 9. The outer radius of the Coupling R_{2}
Figure 10. Constraint g_{2}
Figure 11. Volume minimization
Table 4. The inner radius R_{1}
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 R_{2}
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 g_{1}
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 |
6.1 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.
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.
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:
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.
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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