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Six engineering challenges have been selected for this part for the purpose to assess how well CWXSCSO performs when used to engineering optimization problems. The sine and cosine optimization algorithm (SCA)40, frost and ice optimization algorithm (RIME)41, butterfly optimization algorithm (BOA)42, Harris Eagle Optimization algorithm (HHO)8, and Osprey optimization algorithm (OOA)43 were chosen as the primary three technical applications. The whale optimization algorithm (WOA)7, the locust optimization algorithm (GOA)44, the gray wolf optimization algorithm (GWO)45, the marine predator optimization algorithm (MPA)46, and the frost and ice optimization algorithm (RIME) were used to compare the final three technical applications. Every algorithm in the experiment has a population of 30 and an upper limit of 1000 iterations.

The performance of the modified algorithm pair gets assessed using pressure vessel design issues in this research. The main objective of the pressure vessel design challenge is to decrease the production expenses associated with the pressure vessel. This problem contains the selection of four optimization variables, namely shell thickness ({T}_{S}), head thickness (({T}_{h})), inner radius ((R)), and length of cylinder section without head ((L)). The mathematical description of the pressure vessel design problem is as follows:

variable:

$$overrightarrow{x}=left[{x}_{1} {x}_{2} {x}_{3} {x}_{4}right]=left[{T}_{S} {T}_{h} R Lright]$$

Function:

$$fleft(overrightarrow{x}right)=0.6224{x}_{1}{x}_{3}{x}_{4}+1.7781{x}_{2}{x}_{3}^{2}+3.1661{x}_{1}^{2}{x}_{4}+19.84{x}_{1}^{2}{x}_{3}$$

Constraint condition:

$${g}_{1}left(overrightarrow{x}right)=-{x}_{1}+0.0193{x}_{3}le 0$$

$${g}_{2}left(overrightarrow{x}right)=-{x}_{3}+0.00954{x}_{3}le 0$$

$${g}_{3}left(overrightarrow{x}right)=-pi {x}_{3}^{2}-frac{4}{3}pi {x}_{3}^{3}+1296000le 0$$

$${g}_{4}left(overrightarrow{x}right)={x}_{4}-240le 0$$

Variable interval:

$$0le {x}_{1},{x}_{2}le 99, 10le {x}_{3},{x}_{4}le 200$$

The experimental findings of CWXSCSO and the comparison algorithm are presented in Table 7. The CWXSCSO yields a value of 5886.05. When compared to alternative algorithms, this particular algorithm exhibits a superior competitive advantage in terms of maintaining the proper functioning of the pressure vessel while simultaneously minimizing costs. Benefits in guaranteeing the operation of the pressure vessel while reducing expenses. The updated method demonstrates rapid convergence to the ideal value with the best convergence accuracy, as depicted in Fig.5. In turn, the CWXSCSO facility exhibits exceptional engineering optimization capabilities.

Optimization convergence diagram of pressure vessel design problem.

The issue at hand is the Welded Beam Design (WBD), which involves the utilization of an optimization method to minimize the production cost associated with the design. The optimization problem can be boiled down to the identification of four design variables that meet the constraints of shear stress ((tau )), bending stress ((theta )), beam bending load (left({P}_{c}right)), end deviation ((delta )), and boundary conditions, namely beam length ((l)), height ((t)), thickness ((b)), and weld thickness ((h)). The objective is to minimize the manufacturing cost of welded beams. The problem of welded beams is a common example of a nonlinear programming problem. The mathematical description of the welded beam design problem is as follows:

Variable:

$$overrightarrow{x}=left[{x}_{1} {x}_{2} {x}_{3} {x}_{4}right]=left[h l t bright]$$

Function:

$$fleft(overrightarrow{x}right)=1.10471{x}_{1}^{2}{x}_{2}+0.04811{x}_{3}{x}_{4}left(14.0+{x}_{2}right)$$

Constraint condition:

$${g}_{1}left(overrightarrow{x}right)=tau left(overrightarrow{x}right)-{tau }_{max}le 0$$

$${g}_{2}left(overrightarrow{x}right)=sigma left(overrightarrow{x}right)-{sigma }_{max}le 0$$

$${g}_{3}left(overrightarrow{x}right)=delta left(overrightarrow{x}right)-{delta }_{max}le 0$$

$${g}_{4}left(overrightarrow{x}right)={x}_{1}-{x}_{4}le 0$$

$${g}_{5}left(overrightarrow{x}right)=P-{P}_{c}left(overrightarrow{x}right)le 0$$

$${g}_{6}left(overrightarrow{x}right)=0.125-{x}_{1}le 0$$

$${g}_{7}left(overrightarrow{x}right)=1.10471{x}_{1}^{2}{x}_{2}+0.04811{x}_{3}{x}_{4}left(14.0+{x}_{2}right)-5.0le 0$$

Variable interval:

$$0.1le {x}_{1}le 2,{ 0.1le x}_{2}le 10, 0.1le {x}_{3}le 10 ,{0.1le x}_{4}le 2$$

As can be seen from Table 8, the final result of CWXSCSO is 1.6935. As can be seen in Fig.6, the initial fitness value of the improved algorithm is already very good, and there are several subtle turns later, indicating that it has the ability to jump out of the local optimal. The improved algorithm achieves the purpose of reducing the manufacturing cost, and the cost of manufacturing welded beams is minimal compared with other algorithms.

Optimization convergence diagram of welding beam design problem.

The reducer holds an important place within mechanical systems as a crucial component of the gear box, serving a diverse range of applications. The primary aim of this challenge is to diminish the overall weight of the reducer through the optimization of the seven parameter variables. They are the tooth surface width (b)(=({x}_{1})), the gear module (m(={x}_{2})), the tooth count in the pinion (z(={x}_{3})), the measurement of the initial shaft distance between bearings. ({l}_{1}(={x}_{4})), the distance between the bearings of the second shaft ({l}_{2}(={x}_{5})), the diameter of the initial shaft ({d}_{1}(={x}_{6})) and the measurement of the diameter of the second shaft ({d}_{2}(={x}_{7})). The mathematical description of the speed reducer design problem is as follows:

Variable:

$$overrightarrow{x}=left[{x}_{1} {x}_{2} {x}_{3} {x}_{4} {x}_{5} {x}_{6} {x}_{7}right]=left[b m z {l}_{1} {l}_{2} {d}_{1} {d}_{2}right]$$

Function:

$$fleft(overrightarrow{x}right)=0.7854{x}_{1}{x}_{2}^{2}left(3.3333{x}_{3}^{2}+14.9334{x}_{3}-43.0934right)-1.508{x}_{1}left({x}_{6}^{2}+{x}_{7}^{2}right)+7.4777left({x}_{6}^{3}+{x}_{7}^{3}right)+0.7854({x}_{4}{x}_{6}^{2}+{{x}_{5}x}_{7}^{2})$$

Constraint condition:

$${g}_{1}left(overrightarrow{x}right)=frac{27}{{x}_{1}{x}_{2}^{2}{x}_{3}}-1le 0$$

$${g}_{2}left(overrightarrow{x}right)=frac{397.5}{{x}_{1}{x}_{2}^{2}{x}_{3}^{2}}-1le 0$$

$${g}_{3}left(overrightarrow{x}right)=frac{1.93{x}_{4}^{3}}{{x}_{2}{x}_{3}{x}_{6}^{4}}-1le 0$$

$${g}_{4}left(overrightarrow{x}right)=frac{1.93{x}_{5}^{3}}{{x}_{2}{x}_{3}{x}_{7}^{4}}-1le 0$$

$${g}_{5}left(overrightarrow{x}right)=frac{sqrt{{left(frac{745{x}_{4}}{{x}_{2}{x}_{3}}right)}^{2}+16.9times {10}^{6}}}{110.0{x}_{6}^{3}}-1le 0$$

$${g}_{6}left(overrightarrow{x}right)=frac{sqrt{{left(frac{745{x}_{4}}{{x}_{2}{x}_{3}}right)}^{2}+157.5times {10}^{6}}}{85.0{x}_{6}^{3}}-1le 0$$

$${g}_{7}left(overrightarrow{x}right)=frac{{x}_{2}{x}_{3}}{40}-1le 0$$

$${g}_{8}left(overrightarrow{x}right)=frac{{5x}_{2}}{{x}_{1}}-1le 0$$

$${g}_{9}left(overrightarrow{x}right)=frac{{x}_{1}}{{12x}_{2}}-1le 0$$

$${g}_{10}left(overrightarrow{x}right)=frac{{1.5x}_{6}+1.9}{{x}_{4}}-1le 0$$

$${g}_{11}left(overrightarrow{x}right)=frac{{1.1x}_{7}+1.9}{{x}_{5}}-1le 0$$

Variable interval:

$$2.6le {x}_{1}le 3.6 ,0.7le {x}_{2}le 0.8 ,17le {x}_{3}le 28 , 7.3le {x}_{4}le 8.3 ,7.8le {x}_{5}le 8.3,$$

$$2.9le {x}_{6}le 3.9, 5.0le {x}_{7}le 5.5$$

Table 9 and Fig.7 demonstrate that the modified method is adept at minimizing the weight of the reducer under 11 boundaries. It suggests that the enhancement is effective and may be more effectively utilized in mechanical systems.

Reducer design optimization convergence curve.

This engineering project aims to create a 4-step cone pulley with a minimal weight by looking at 5 design elements. Four variables represent the diameter of individual step of the pulley, denoted as ({d}_{i}(i=mathrm{1,2},mathrm{3,4})), while the final variable represents the magnitude of the pulley's breadth, denoted as (w). There are 8 nonlinear constraints and 3 linear constraints in the problem. The restriction is to maintain uniformity in the belt length ({C}_{i}), tension ratio ({R}_{i}), and belt transfer power ({P}_{i}) throughout all steps. The mathematical description of the step cone pulley problem is as follows:

Function:

$$fleft(xright)=rho omega left[{d}_{1}^{2}left{1+{left(frac{{N}_{1}}{N}right)}^{2}right}+{d}_{2}^{2}left{1+{left(frac{{N}_{2}}{N}right)}^{2}right}+{d}_{3}^{2}left{1+{left(frac{{N}_{3}}{N}right)}^{2}right}+{d}_{4}^{2}left{1+{left(frac{{N}_{4}}{N}right)}^{2}right}right]$$

Constraint condition:

$${h}_{1}left(xright)={C}_{1}-{C}_{2}=0, {h}_{2}left(xright)={C}_{1}-{C}_{3}=0 , {h}_{3}left(xright)={C}_{1}-{C}_{4}=0$$

$${g}_{mathrm{1,2},mathrm{3,4}}left(xright)={R}_{i}ge 2, {g}_{mathrm{5,6},mathrm{7,8}}left(xright)={P}_{i}ge left(0.75*745.6998right)$$

where:

$${C}_{i}=frac{pi {d}_{i}}{2}left(1+frac{{N}_{i}}{N}right)+frac{{left(frac{{N}_{i}}{N}-1right)}^{2}}{4a}+2a i=left(mathrm{1,2},mathrm{3,4}right)$$

$${R}_{i}=expleft[mu left{pi -2{{text{sin}}}^{-1}left{left(frac{{N}_{i}}{N}-1right)frac{{d}_{i}}{2a}right}right}right] i=left(mathrm{1,2},mathrm{3,4}right)$$

$${P}_{i}=stwleft[1-expleft[-mu left{pi -2{{text{sin}}}^{-1}left{left(frac{{N}_{i}}{N}-1right)frac{{d}_{i}}{2a}right}right}right]right]frac{pi {d}_{i}{N}_{i}}{60} i=left(mathrm{1,2},mathrm{3,4}right)$$

$$rho =7200kg/{m}^{3} , a=3m ,mu =0.35 ,s=1.75MPa ,t=8mm$$

Variable interval:

$$0le {d}_{1}, {d}_{2}le 60, 0le {d}_{3}, omega le 90$$

Table 10 clearly demonstrates that the MPA method outperforms the CWXSCSO algorithm, but it still possesses certain advantages over other algorithms. Figure8 illustrates that while the precision of convergence in CWXSCSO is less than that of MPA, its convergence speed beats that of MPA. Despite lacking MPA for the stepping cone pulley problem, CWXSCSO still has the benefit of rapid convergence speed.

Optimization convergence diagram of step cone pulley problem.

In power mechanical systems, the design of a planetary gear train presents a limited optimization problem. The issue encompasses three optimization variables, specifically the quantity of gear teeth (left({N}_{1},{N}_{2},{N}_{3},{N}_{4},{N}_{5},{N}_{6}right)), gear modulus (left({m}_{1},{m}_{2}right)), and the figure of merit (left(pright)). The primary aim of the issue is to limit the maximum error associated with the transmission ratio employed in automotive production. The issue at hand encompasses a total of six integer variables, three discrete variables, and eleven distinct geometric and assembly restrictions. The mathematical description of the planetary gear train design optimization problem is as follows:

Variable:

$$x=left({x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{6},{x}_{7},{x}_{8},{x}_{9}right)=left({N}_{1},{N}_{2},{N}_{3},{N}_{4},{N}_{5},{N}_{6},{m}_{1},{m}_{2},pright)$$

Function:

$$fleft(xright)=maxleft|{i}_{k}-{i}_{ok}right|, k=left{mathrm{1,2},dots ,Rright}$$

where:

$${i}_{1}=frac{{N}_{6}}{{N}_{4}}, {i}_{o1}=3.11, {i}_{2}=frac{{N}_{6}left({N}_{1}{N}_{3}+{N}_{2}{N}_{4}right)}{{N}_{1}{N}_{3}left({N}_{6}+{N}_{4}right)}, {i}_{OR}=-3.11, {I}_{R}=-frac{{N}_{2}{N}_{6}}{{N}_{1}{N}_{3}}, {i}_{O2}=1.84$$

Constraint condition:

$${g}_{1}left(xright)={m}_{2}left({N}_{6}+2.5right)-{D}_{max}le 0$$

$${g}_{2}left(xright)={m}_{1}left({N}_{1}+{N}_{2}right)+{m}_{1}left({N}_{2}+2right)-{D}_{max}le 0$$

$${g}_{3}left(xright)={m}_{2}left({N}_{4}+{N}_{5}right)+{m}_{2}left({N}_{5}+2right)-{D}_{max}le 0$$

$${g}_{4}left(xright)=left|{m}_{1}left({N}_{1}+{N}_{2}right)-{m}_{1}left({N}_{6}+{N}_{3}right)right|-{m}_{1}-{m}_{2}le 0$$

$${g}_{5}left(xright)=-left({N}_{1}+{N}_{2}right){text{sin}}left(frac{pi }{p}right)+{N}_{2}+2+{delta }_{22}le 0$$

$${g}_{6}left(xright)=-left({N}_{6}-{N}_{3}right){text{sin}}left(frac{pi }{p}right)+{N}_{3}+2+{delta }_{33}le 0$$

$${g}_{7}left(xright)=-left({N}_{4}+{N}_{5}right){text{sin}}left(frac{pi }{p}right)+{N}_{5}+2+{delta }_{55}le 0$$

$${g}_{8}left(xright)={left({N}_{3}+{N}_{5}+2+{delta }_{35}right)}^{2}-{left({N}_{6}-{N}_{3}right)}^{2}-{left({N}_{4}+{N}_{5}right)}^{2}+2left({N}_{6}-{N}_{3}right)left({N}_{4}+{N}_{5}right){text{cos}}left(frac{2pi }{p}-beta right)le 0$$

$${g}_{9}left(xright)={N}_{4}-{N}_{6}+{2N}_{5}+2{delta }_{56}+4le 0$$

$${g}_{10}left(xright)={2N}_{3}-{N}_{6}+{N}_{4}+2{delta }_{34}+4le 0$$

$${h}_{1}left(xright)=frac{{N}_{6}-{N}_{4}}{p}=integer$$

where:

$${delta }_{22}={delta }_{33}={delta }_{55}={delta }_{35}={delta }_{56}=0.5$$

$$beta =frac{{cos}^{-1}left({left({N}_{4}+{N}_{5}right)}^{2}+{left({N}_{6}-{N}_{3}right)}^{2}-{left({N}_{3}+{N}_{5}right)}^{2}right)}{2left({N}_{6}-{N}_{3}right)left({N}_{4}+{N}_{5}right)}$$

Variable interval:

$$P=left(mathrm{3,4},5right), {m}_{1},{m}_{2}=left(mathrm{1.75,2.0,2.25,2.5,2.75,3.0}right) , 17le {N}_{1}le 96,$$

$$14le {N}_{2}le 54, 14le {N}_{3}le 51, 17le {N}_{4}le 46, 14le {N}_{5}le 51, 48le {N}_{6}le 124$$

Based on the data shown in Fig.9 and Table 11, it is evident that CWXSCSO continues to outperform other methods in terms of convergence accuracy and convergence speed. This illustrates the potential for widespread implementation and utilization of the upgraded algorithm in power machinery.

Convergence curve of planetary gear train design optimization problem.

The issue of robot hand claws is a complex challenge within the field of mechanical structure engineering. The goal of the robot clamping optimization is to minimize the disparity between the highest and lowest magnitudes of forces. The challenge of robot grippers encompasses a total of seven continuous design variables the three connecting rods ((a,b,c)), the vertical displacement of the linkages ((d)), the vertical distance separating the initial node of the robotic arm from the end of the actuator ((e)), the displacement in the horizontal direction between the actuator end and the linkages node ((f)), and the angle of the second and third linkages in a geometric context (left(rho right)). There appear a total of seven distinct limitations. The mathematical description of the robot clamping optimization problem is as follows:

Variable:

$$x=left({x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{6},{x}_{7}right)=left(a,b,c,d,e,f,pright)$$

Function:

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