# An Optimal Design Method for Compliant Mechanisms

Authors: Ngoc Le Chau; Ngoc Thoai Tran; Thanh-Phong Dao
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

Compliant mechanisms are crucial parts in precise engineering but modeling techniques are restricted by a high complexity of their mechanical behaviors. Therefore, this paper devotes an optimal design method for compliant mechanisms. The integration method is a hybridization of statistics, finite element method, artificial intelligence, and metaheuristics. In order to demonstrate the superiority of the method, one degree of freedom is considered as a study object. Firstly, numerical datasets are achieved by the finite element method. Subsequently, the main design parameters of the mechanism are identified via analysis of variance. Desirability of both displacement and frequency of the mechanism is determined, and then, they are embedded inside a fuzzy logic system to combine into a single fitness function. Then, the relationship between the fine design variables and the fitness function is modeled using the adaptive network-based fuzzy inference system. Next, the single fitness function is maximized via moth-flame optimization algorithm. The optimal results determined that the frequency is 79.517 Hz and displacement is 1.897 mm. In terms of determining the global optimum solution, the current method is compared with the Taguchi, desirability, and Taguchi-integrated fuzzy methods. The results showed that the current method is better than those methods. Additionally, the devoted method outperforms the other metaheuristic algorithms such as TLBO, Jaya, PSOGSA, SCA, ALO, and LAPO in terms of faster convergence. The result of this study will be considered to apply for multiple-degrees-of-freedom compliant mechanisms in future work.


---

## Body

## 1. Introduction

Compliant mechanisms are specific mechanical devices, the mobility of which is inherently based on elastic energy [1–4]. Owing to the emerging strengths, compliant mechanisms have been receiving a great interest in industrial applications, for example, gripper [5], printing [6], nanopositioner [7], constant force mechanism [8], multistable equilibrium positions [9], microelectromechanical systems [10], precision diamond turning [11], and energy harvesting [12]. Unlike rigid-body mechanisms, compliant mechanisms gain the excellent benefits such as a monolithic structure, lightweight, free friction, and free lubricant. On the other hand, rigid-body counterparts make friction and clearance thanks to kinematic joints such as revolute, prismatic, cylindrical, and spherical bearings or gears; meanwhile compliant mechanism gains smooth motions. By using rigid links and kinematic joints, rigid-body counterparts are easily analyzed through the traditional machines and mechanism theory [13]. On the contrary, theory for analyzing and synthesizing compliant mechanisms has been facing difficulties thanks to simultaneous coupling of kinematic and mechanical behaviors. Until now, a lot of different approaches for modeling compliant mechanism have been suggested, for example, pseudo-rigid-body model [14, 15], Castigliano [16], compliance [17], beam theory [18], dynamic stiffness [19], empirical technique [20], constraint-beam model [21], Euler-Bernoulli [22], and finite element method (FEM) [23]. In comparison with the mentioned methods, FEM is a useful tool for solving highly nonlinear problems.To fulfill the gap between previous studies and the present study, an optimal design method is suggested. One-degree-of-freedom (DOF) compliant mechanism is employed as a study example to demonstrate the method’s effectiveness. Regarding a fast tool servo [11], the 1-DOF requires a large range of displacement and a highly natural frequency. Besides, a small parasitic motion and stress are considered as two important constraints. Nowadays, optimization problem for compliant mechanisms can be classified into three main areas: topology optimization [24, 25], shape optimization, and size optimization [22, 26, 27]. In the present article, a multiobjective optimization (MOO) for the 1-DOF mechanism is proposed to improve its responses. Previously, there have been a few studies to deal with the MOO for compliant mechanisms but efficiency of techniques is still a challenge [28–30].Moreover, there have been different types of optimization methods to deal with an optimization process. In general, a mathematical model is created before implementing a MOO problem. However, the analytical approaches have not been suitable for complex structures. In such a circumstance, data-based approaches are promising tools, which can predict and optimize the performances simultaneously. In order to save manufacturing costs, this article proposes a combination of numerical simulation, statistical techniques, and metaheuristics in terms of a reliable and global solution. Several approaches can be summarized as Taguchi [31], desirability [32], grey relation [33], and Taguchi-fuzzy (TF) [34] but most of them may reach a local solution. On the contrary, in order to achieve a global value, surrogate model is coupled with metaheuristics. The surrogate models include response surface approach [35, 36], Kriging [37], neural network [38], fuzzy [39], and adaptive-network-based fuzzy inference system [40]. Among them, adaptive-network-based fuzzy inference system (ANFIS) is an exact predictor. Related to metaheuristics, a variety of different algorithms were proposed, such as genetic algorithm [41], particle swarm optimization [42], and cuckoo search [43]. These metaheuristics require tuned parameters, such as teaching-learning-based algorithm [44], Jaya algorithm [45], and lightning attachment procedure optimization [46]. However, these algorithms are still limited by a low convergence speed. Then, other metaheuristics have been proposed to speed up the convergent time, for example, moth-flame optimization [47], ant lion optimizer [48], particle swarm optimization-based gravitational search algorithm [49], and sine-cosine algorithm [50]. In the present article, the moth-flame optimization is chosen for the 1-DOF mechanism due to its fast convergence.The present paper aims to contribute an optimal design method for compliant mechanisms. The method undergoes six phases: Firstly, the nonlinear FEM is utilized to analyze the aforementioned behaviors of the 1-DOF mechanism. Secondly, some new populations for moth-flame optimization are discovered by investigating the sensitivity of parameters. Thirdly, real values of objective functions are converted into the desirability to suppress influences of units. Subsequently, fuzzy logic system is developed to bring all desirabilities into a single fitness function. The fitness function is defined as a combined objective function of multiple performances of the 1-DOF mechanism. Then, the relationship among the fine geometrical parameters and the established single fitness function is formulated through ANFIS model. Lastly, the single fitness function is then maximized via the moth-flame optimization. The remainder of this paper is organized as follows: Section2 presents the computational method. A mechanical design of the 1-DOF mechanism is provided in Section 3. Practical implications and discussion are analyzed in Section 4. Conclusions are given in Section 5.


## 2. Optimal Design Method

In order to resolve a MOO for a 1-DOF mechanism, an optimal design framework is given (see in Figure1). The optimization procedure is summarized by the following stages.Figure 1
Flowchart of the optimal design method for compliant mechanisms.


### 2.1. Stage 1: Initial Design

In the first stage, a draft model of the one-DOF mechanism is created. In this study, the computational method is offered to resolve MOO design for the 1-DOF mechanism. Numerical examples are investigated involving the usefulness of the developed computational method. The optimization process undergoes the following stepwise procedure.Design Description. The 1-DOF compliant mechanism should achieve whole good frequency and displacement. Additionally, a small stress and parasitic motion are two constraints. In other words, a frequency aims to increase the response of system. A large displacement is expected to enlarge the range of positioning.Design Variables. Geometrical parameters are identified as main design variables for 1-DOF compliant mechanism.Objective Functions. Two design objectives include the frequency and the displacement.3D Model. A 3D model is initialized, and then the frequency, displacement, parasitic motion, and stress are retrieved by finite element analysis (FEA) simulations.Numerically Experimental Design. Numerical experiments are laid out by Box-Behnken design (BBD).Numerical Dataset. Numerical datasets are retrieved through simulations.Investigation of Sensitivity. Sensitivity analysis of all design variables is to identify several key parameters directly affecting both objective functions. On the other hand, this step determines design variables again and eliminates some low-significance parameters through analysis of variance (ANOVA).Refined Design Variables. Main design variables are determined again to prepare several new population spaces for moth-flame optimization algorithm.Redesigned Numerical Experiments. Numerical experiments are built again based on the refined design variables.Update 3D Model. 3D model is updated corresponding to a new population. Subsequently, numerical data are retrieved through FEA simulations.


### 2.2. Stage 2: Computation of Desirability Value

The aim of calculating the desirability is to suppress influences of different units among the frequency (Hz) and the displacement (mm). This stage undergoes some substeps as below.Update Numerical Data. Numerical data are retrieved again based on the refined design variables.Desirability Value. The desirability was utilized as a predictor. In this paper, the exponential type is used. The larger-the-better type is used for both objective functions in this article.The larger-the-better type is explained as(1)Di=0,F∗≤BL,Di=F∗−BLBU−BLr,BL≤F∗≤BU,Di=1,F∗≥BU,where the desirability is denoted by Di. The ith objective function is denoted by F∗. BL and BU are lower range and upper range of F∗, respectively. R is desirability function index.


### 2.3. Stage 3: Modeling by the Fuzzy Logic System

The fuzzy logic system [51, 52] is employed to change both desirabilities into a single fitness function. Then, fuzzy inference system (FIS) is then utilized to generate a multiperformance characteristic index (MPCI) or the so-called single fitness function. This system is illustrated as shown in Figure 1.


### 2.4. Stage 4: Modeling by ANFIS

ANFIS is a popular technique by combining ANN and FIS [53]. The purpose of ANFIS model is to model the relations among the refined design variables and objective functions (see Figure 1). In this paper, datasets are divided into 70% for training and 30% for testing. Performances indexes, root mean square error (RMSE), and correlation coefficient (R2) are utilized to evaluate the predictor.


### 2.5. Stage 5: Optimization Algorithm

Moth-flame algorithm (MFO) is mimicked by moth’s behavior [47]. As moths see the light source, they fly in a spiral path. MFO has been successfully applied for many engineering areas thanks to simple usage and fast convergence rate. In this paper, MFO is extended to reach a global optimal design for the 1-DOF mechanism. A maximum termination criterion is chosen as 105 in this study. More details of the MFO can be found in literature [47]. A flowchart of MFO is shown in Figure 1.


## 2.1. Stage 1: Initial Design

In the first stage, a draft model of the one-DOF mechanism is created. In this study, the computational method is offered to resolve MOO design for the 1-DOF mechanism. Numerical examples are investigated involving the usefulness of the developed computational method. The optimization process undergoes the following stepwise procedure.Design Description. The 1-DOF compliant mechanism should achieve whole good frequency and displacement. Additionally, a small stress and parasitic motion are two constraints. In other words, a frequency aims to increase the response of system. A large displacement is expected to enlarge the range of positioning.Design Variables. Geometrical parameters are identified as main design variables for 1-DOF compliant mechanism.Objective Functions. Two design objectives include the frequency and the displacement.3D Model. A 3D model is initialized, and then the frequency, displacement, parasitic motion, and stress are retrieved by finite element analysis (FEA) simulations.Numerically Experimental Design. Numerical experiments are laid out by Box-Behnken design (BBD).Numerical Dataset. Numerical datasets are retrieved through simulations.Investigation of Sensitivity. Sensitivity analysis of all design variables is to identify several key parameters directly affecting both objective functions. On the other hand, this step determines design variables again and eliminates some low-significance parameters through analysis of variance (ANOVA).Refined Design Variables. Main design variables are determined again to prepare several new population spaces for moth-flame optimization algorithm.Redesigned Numerical Experiments. Numerical experiments are built again based on the refined design variables.Update 3D Model. 3D model is updated corresponding to a new population. Subsequently, numerical data are retrieved through FEA simulations.


## 2.2. Stage 2: Computation of Desirability Value

The aim of calculating the desirability is to suppress influences of different units among the frequency (Hz) and the displacement (mm). This stage undergoes some substeps as below.Update Numerical Data. Numerical data are retrieved again based on the refined design variables.Desirability Value. The desirability was utilized as a predictor. In this paper, the exponential type is used. The larger-the-better type is used for both objective functions in this article.The larger-the-better type is explained as(1)Di=0,F∗≤BL,Di=F∗−BLBU−BLr,BL≤F∗≤BU,Di=1,F∗≥BU,where the desirability is denoted by Di. The ith objective function is denoted by F∗. BL and BU are lower range and upper range of F∗, respectively. R is desirability function index.


## 2.3. Stage 3: Modeling by the Fuzzy Logic System

The fuzzy logic system [51, 52] is employed to change both desirabilities into a single fitness function. Then, fuzzy inference system (FIS) is then utilized to generate a multiperformance characteristic index (MPCI) or the so-called single fitness function. This system is illustrated as shown in Figure 1.


## 2.4. Stage 4: Modeling by ANFIS

ANFIS is a popular technique by combining ANN and FIS [53]. The purpose of ANFIS model is to model the relations among the refined design variables and objective functions (see Figure 1). In this paper, datasets are divided into 70% for training and 30% for testing. Performances indexes, root mean square error (RMSE), and correlation coefficient (R2) are utilized to evaluate the predictor.


## 2.5. Stage 5: Optimization Algorithm

Moth-flame algorithm (MFO) is mimicked by moth’s behavior [47]. As moths see the light source, they fly in a spiral path. MFO has been successfully applied for many engineering areas thanks to simple usage and fast convergence rate. In this paper, MFO is extended to reach a global optimal design for the 1-DOF mechanism. A maximum termination criterion is chosen as 105 in this study. More details of the MFO can be found in literature [47]. A flowchart of MFO is shown in Figure 1.


## 3. Numerical Study

The proposed 1-DOF mechanism is a potential positioner for precision system. This mechanism is expected to be used for a fast tool servo system whose applications can be found in the literatures [54–56]. In earlier design phase, in order to decrease the cost of a real fabrication process, the present study suggests a numerical optimization method for the 1-DOF mechanism.


### 3.1. Design Description

Figures2(a) and 2(b) show 2D and 3D diagrams of 1-DOF compliant mechanism. In the middle, the mechanism is fixed holes. The mechanism includes three flexure hinges, named as FH-1, FH-2, and FH-3. Such FHs are connected through rigid links. The mechanism includes an input end (input load F of 25 N from an actuator) and an output is used to fix a cutting tool. The output displacement moves in vertical direction. At the same time, it also moves in horizontal direction, so-called parasitic motion. Flexure hinges with rectangular cross section permit a large displacement but this is a monolithic structure. Because it is subset of compliant mechanism and works in an elastic limitation of material, its motions are largely dependent on cross section of FHs; therefore, this article optimizes geometrical parameters of FHs. Some significant parameters of the proposed mechanism consist of dimensions of FHs [T1, L1, T2, L2, T3, and L3]. Remaining parameters (L, W, and H) are assigned as constant values. The 1-DOF mechanism is proposed for fast tool servo-assisted diamond cutting system to produce fined microstructure surfaces. Table 1 gives parameters of the mechanism. This mechanism is made by material Al 7075 with yield strength of 503 MPa.Figure 2
(a) 2D diagram and (b) 3D model (unit: mm).
(a)(b)Table 1
Parameters.
SymbolValueT10.72mm≤T1≤1.04mmL114.5mm≤L1≤16.5mmT20.45mm≤T2≤0.65mmL222.5≤L2≤27.5T30.54mm≤T3≤0.78mmL318mm≤L3≤22mmW10 mmH50 mmL120 mm


### 3.2. Numerical Simulation

From Figure2, a load of 25 N is employed to achieve the output response. Flexure hinges are refined two times to reach a good meshing. Solid 186 type of elements is used. The results determined that there are 747 elements and 5208 nodes, as depicted in Figure 3(a). In order to reach a better accuracy of simulation results, Skewness criteria are employed. The meshing result found that the value of this performance metric is about 0.40684, and this shows a good meshing quality (see Figure 3(b)).Figure 3
Simulation setup: (a) meshing process; (b) metric distribution of mesh.
(a)(b)


### 3.3. Optimization Statement

As discussed above, the 1-DOF compliant mechanism is considered for translational manipulators where a high natural frequency over 70 Hz, a large displacement over 1.7 mm, a minimal parasitic motion under 0.02 mm, and a good strength are required. The optimization statement is described.Findx = [T1, L1, T2, L2, T3, L3]T.(2)Maximizef1x,(3)Maximixef2x.They are subject to constraints:(4)f1x≥70Hz,f2x≥1.7mm,f3x≤0.02mm,f4x≤503MPan.Initial space of design variables (unit: mm) is(5)0.72≤T1≤1.04,14.5≤L1≤16.5,0.45≤T2≤0.65,22.5≤L2≤27.5,0.54≤T3≤0.78,18≤L3≤22.f1 (x), f2 (x), f3 (x), and f4 (x) represent natural frequency, displacement, parasitic error, and stress correspondingly. N is safety factor. The ranges of parameters are determined based on the experiences in the field and further fabrication capacity of devices.


## 3.1. Design Description

Figures2(a) and 2(b) show 2D and 3D diagrams of 1-DOF compliant mechanism. In the middle, the mechanism is fixed holes. The mechanism includes three flexure hinges, named as FH-1, FH-2, and FH-3. Such FHs are connected through rigid links. The mechanism includes an input end (input load F of 25 N from an actuator) and an output is used to fix a cutting tool. The output displacement moves in vertical direction. At the same time, it also moves in horizontal direction, so-called parasitic motion. Flexure hinges with rectangular cross section permit a large displacement but this is a monolithic structure. Because it is subset of compliant mechanism and works in an elastic limitation of material, its motions are largely dependent on cross section of FHs; therefore, this article optimizes geometrical parameters of FHs. Some significant parameters of the proposed mechanism consist of dimensions of FHs [T1, L1, T2, L2, T3, and L3]. Remaining parameters (L, W, and H) are assigned as constant values. The 1-DOF mechanism is proposed for fast tool servo-assisted diamond cutting system to produce fined microstructure surfaces. Table 1 gives parameters of the mechanism. This mechanism is made by material Al 7075 with yield strength of 503 MPa.Figure 2
(a) 2D diagram and (b) 3D model (unit: mm).
(a)(b)Table 1
Parameters.
SymbolValueT10.72mm≤T1≤1.04mmL114.5mm≤L1≤16.5mmT20.45mm≤T2≤0.65mmL222.5≤L2≤27.5T30.54mm≤T3≤0.78mmL318mm≤L3≤22mmW10 mmH50 mmL120 mm


## 3.2. Numerical Simulation

From Figure2, a load of 25 N is employed to achieve the output response. Flexure hinges are refined two times to reach a good meshing. Solid 186 type of elements is used. The results determined that there are 747 elements and 5208 nodes, as depicted in Figure 3(a). In order to reach a better accuracy of simulation results, Skewness criteria are employed. The meshing result found that the value of this performance metric is about 0.40684, and this shows a good meshing quality (see Figure 3(b)).Figure 3
Simulation setup: (a) meshing process; (b) metric distribution of mesh.
(a)(b)


## 3.3. Optimization Statement

As discussed above, the 1-DOF compliant mechanism is considered for translational manipulators where a high natural frequency over 70 Hz, a large displacement over 1.7 mm, a minimal parasitic motion under 0.02 mm, and a good strength are required. The optimization statement is described.Findx = [T1, L1, T2, L2, T3, L3]T.(2)Maximizef1x,(3)Maximixef2x.They are subject to constraints:(4)f1x≥70Hz,f2x≥1.7mm,f3x≤0.02mm,f4x≤503MPan.Initial space of design variables (unit: mm) is(5)0.72≤T1≤1.04,14.5≤L1≤16.5,0.45≤T2≤0.65,22.5≤L2≤27.5,0.54≤T3≤0.78,18≤L3≤22.f1 (x), f2 (x), f3 (x), and f4 (x) represent natural frequency, displacement, parasitic error, and stress correspondingly. N is safety factor. The ranges of parameters are determined based on the experiences in the field and further fabrication capacity of devices.


## 4. Practical Implications and Discussion

### 4.1. Determination of Main Parameters

Involving whole initial design variables, design of numerical experiments is built by BBD technique. Table2 gives initial design parameters and their ranges. Each experiment is implemented by simulations. The initial results are retrieved in Table 3.Table 2
Initial design variables and range (unit: mm).
NotationRangeLevel 1Level 2Level 3T10.72≤T1≤1.040.720.881.04L114.5≤L1≤16.514.515.516.5T20.45≤T2≤0.650.450.550.65L222.5≤L2≤27.522.524.527.5T30.54≤T3≤0.780.540.660.78L318≤L3≤22182022Table 3
Initial results.
TrialParameters (unit: mm)Frequency (Hz)Displacement (mm)Parasitic motion (mm)Stress (MPa)T1L1T2L2T3L3f1 (x)f2 (x)f3 (x)f4 (x)10.88150.55250.6620103.99160.93350.0124162.7020.7213.50.5522.50.6620112.93610.77610.0110131.6931.0413.50.5522.50.6620112.53400.77870.0112134.3840.7216.50.5522.50.6620113.42980.76630.0106130.9851.0416.50.5522.50.6620112.36020.77830.0111134.1260.7213.50.5527.50.662091.03061.23350.0158151.3071.0413.50.5527.50.662090.57611.24750.0165167.1780.7216.50.5527.50.662092.12461.20130.0157159.1791.0416.50.5527.50.662091.88951.19850.0157159.29100.8813.50.45250.542087.74641.28450.0130152.76110.8816.50.45250.542085.41901.36010.0139151.35120.8813.50.65250.5420101.81920.87860.0102108.25130.8816.50.65250.5420101.15710.89220.0103106.90140.8813.50.45250.7820102.56910.99770.0128151.42150.8816.50.45250.7820100.47351.03620.0130154.94160.8813.50.65250.7820127.65350.62160.0104107.23170.8816.50.65250.7820129.24420.60420.0101106.92180.88150.4522.50.6618103.20410.98340.0155152.76190.88150.6522.50.6618136.82870.51970.008293.28200.88150.4527.50.661876.05151.76370.0210225.27210.88150.6527.50.6618112.59880.79960.0117109.54220.88150.4522.50.662294.85561.12170.0155152.93230.88150.6522.50.6622117.40830.65270.008087.21240.88150.4527.50.662272.71041.91200.0210227.36250.88150.6527.50.6622102.51060.93220.0119109.20260.72150.5522.50.542098.05390.96190.0110130.67271.04150.5522.50.542098.07510.95510.0109132.22280.72150.5527.50.542085.15011.36710.0153157.48291.04150.5527.50.542085.20191.35840.0156157.87300.72150.5522.50.7820123.76920.66680.0106134.85311.04150.5522.50.7820122.53880.67660.0106136.39320.72150.5527.50.782096.08111.11880.0157160.43331.04150.5527.50.782094.78341.14950.0161170.74340.8813.50.55250.541898.06781.03400.0130168.07350.8816.50.55250.541897.27161.04840.0128155.91360.8813.50.55250.7818112.32990.82370.0129166.04370.8816.50.55250.7818103.99160.82180.0127164.79380.8813.50.55250.5422112.93611.27570.0127165.29390.8816.50.55250.5422112.53401.27790.0139150.35400.8813.50.55250.7822113.42980.91780.0135159.05410.8816.50.55250.7822112.36020.91980.0130166.97420.72150.45250.661891.03061.04120.0130152.59431.04150.45250.661890.57611.08720.0133149.97440.72150.65250.661892.12460.63970.0102106.64451.04150.65250.661891.88950.64040.0102106.75460.72150.45250.662287.74641.20740.0135155.46471.04150.45250.662285.41901.21520.0134151.53480.72150.65250.6622101.81920.77500.0102107.25491.04150.65250.6622111.36040.76150.0100106.38Case study 1 focuses on the sensitivity analysis for the natural frequency. In Table4, the results of ANOVA determined that T1 with contribution of 0.02% and L1 with contribution of 0% and their p values are 0.567 and 0.891, which are larger than 0.05. The contributions of T1 and L1 are very small, and they are therefore deleted from modeling and optimization process. Additionally, a matrix plot is drawn to show effects of all parameters on the frequency. As seen in Figure 4, it also has a similar conclusion in Table 4. To summarize, case study 1 deals with the main design parameters, including T2, L2, T3, and L3.Table 4
The frequency’s ANOVA.
SourceDfSeq SSContributionAdj SSAdj MSF valuep valueModel279662.8898.639662.88357.8855.89≤0.001Linear69110.0892.999110.081518.35237.10≤0.001T112.160.022.162.160.340.567L110.120.000.120.120.020.891T213680.1437.563680.143680.14574.67≤0.001L212715.4427.722715.442715.44424.03≤0.001T312085.9121.292085.912085.91325.72≤0.001L31626.316.39626.31626.3197.80≤0.001Square6250.432.56250.4341.746.520.001T1 ∗ T117.110.0712.2612.261.910.181L1 ∗ L114.490.050.400.400.060.804T2 ∗ T21186.441.9039.1739.176.120.022L2 ∗ L2130.950.3251.6051.608.060.010T3 ∗ T318.500.0919.6419.643.070.095L3 ∗ L3112.950.1312.9512.952.020.1702-Way interaction15302.363.09302.3620.163.150.008T1 ∗ L110.030.000.030.030.000.951T1 ∗ T211.290.011.291.290.200.658T1 ∗ L210.030.000.030.030.010.942T1 ∗ T310.850.010.850.850.130.720T1 ∗ L311.070.011.071.070.170.687L1 ∗ T213.580.043.583.580.560.463L1 ∗ L210.540.010.540.540.090.773L1 ∗ T310.650.010.650.650.100.754L1 ∗ L310.030.000.030.030.000.950T2 ∗ L2112.930.1312.9312.932.020.170T2 ∗ T3172.270.7472.2772.2711.280.003T2 ∗ L3159.820.6159.8259.829.340.006L2 ∗ T31110.011.12110.01110.0117.18≤0.001L2 ∗ L3125.700.2625.7025.704.010.058T3 ∗ L3113.580.1413.5813.582.120.160Error21134.481.37134.486.40Total489797.36100.00Figure 4
Sensitivity plot of the frequency.Case study 2 deals with the sensitivity analysis for displacement. Based on the ANOVA results in Table5, the parameter’s contributions T1, L1, and L3 are very small with 1.57%, 1.33%, and 1.86%, respectively.Table 5
The displacement’s ANOVA.
SourceDfSeq SSContribution (%)Adj SSAdj MSF valuep valueModel275.8298883.265.829880.215923.870.001Linear64.5944265.614.594420.7657413.72≤0.001T110.109931.570.109930.109931.970.175L110.093391.330.093390.093391.670.210T211.6499523.561.649951.6499529.56≤0.001L212.1464430.652.146442.1464438.45≤0.001T310.464726.640.464720.464728.330.009L310.129991.860.129990.129992.330.142Square60.390325.570.390320.065051.170.361T1 ∗ T110.115081.640.045570.045570.820.376L1 ∗ L110.171382.450.059440.059441.060.314T2 ∗ T210.028330.400.000820.000820.010.905L2 ∗ L210.000160.000.027040.027040.480.494T3 ∗ T310.051030.730.075370.075371.350.258L3 ∗ L310.024350.350.024350.024350.440.5162-Way interaction150.8451412.070.845140.056341.010.482T1 ∗ L110.290774.150.290770.290775.210.033T1 ∗ T210.000550.010.000550.000550.010.922T1 ∗ L210.143842.050.143840.143842.580.123T1 ∗ T310.000390.010.000390.000390.010.934T1 ∗ L310.000340.000.000340.000340.010.938L1 ∗ T210.001730.020.001730.001730.030.862L1 ∗ L210.267073.810.267070.267074.780.040L1 ∗ T310.000450.010.000450.000450.010.929L1 ∗ L310.000010.000.000010.000010.000.990T2 ∗ L210.127781.820.127780.127782.290.145T2 ∗ T310.000540.010.000540.000540.010.923T2 ∗ L310.000220.000.000220.000220.000.951L2 ∗ T310.001700.020.001700.001700.030.863L2 ∗ L310.000010.000.000010.000010.000.989T3 ∗ L310.009740.140.009740.009740.170.680Error211.1722116.741.172210.05582Total487.00209100.00The results of Figure5 also have the same conclusion in Table 5. It means that the parameters T1, L1, and L3 should be suppressed during modeling and optimization procedure.Figure 5
Sensitivity plot of the displacement.


### 4.2. Modeling and Optimization

#### 4.2.1. Determination of MFs Types

Table6 assigns the fuzzy variables for MFs. In this article, the frequency and displacement desirabilities are two inputs of the FIS. The MFs types for two inputs and an output of the FIS system are illustrated in Figures 6 and 7.Table 6
Proposed fuzzy variables.
AbbreviationSSRSSSAAALNLLSLFuzzy variableSo smallRelatively smallSmallSmall averageAverageAverage largeNear largeLargeSo largeFigure 6
MFs types: (a) the frequency and (b) the displacement.
(a)(b)Figure 7
MFs type for the output of FIS system.


#### 4.2.2. Investigation on Case Study 1

As discussed in the previous section, overall initial design variables are limited. Those factors actually contribute the responses of 1-DOF mechanism. Besides, spaces of parameters are newly initialized for the modeling and optimization process. The optimization formulation is stated as follows.Findx = [T2, L2, T3, L3]T.(6)Maximizefxcase1,(7)s.t.f1x≥70Hz,f2x≥1.7mm,f3x≤0.02mm,f4x≤503MPan,0.45mm≤T2≤0.65mm,22.5mm≤L2≤27.5mm,0.54mm≤T3≤0.78mm,18mm≤L3≤22mm.f (x) is a single fitness function. Table 7 shows the results for case study 1. Table 8 presents the MFs variables for the fuzzy if-then rules in modeling both objective functions into a single objective function.Table 7
Case study 1: simulated results.
TrialRefined parameters (unit: mm)FrequencyDisplacementParasitic motionStressT2L2T3L3f1 (x) (Hz)f2 (x) (mm)f3 (x) (mm)f4 (x) (MPa)10.55250.6620104.04820.93360.0124162.735520.4522.50.662099.31841.04590.0155153.047230.6522.50.6620127.18880.58060.008192.951640.4527.50.662074.18231.84190.0211227.180350.6527.50.6620107.04810.87420.0120117.824960.55250.541898.74761.01890.0127165.136970.55250.7818112.52950.82210.0127164.978580.55250.542285.98011.26110.0132158.412590.55250.7822105.50510.91830.0129164.3392100.45250.6618100.18991.04110.0130152.5881110.65250.6618124.04860.65190.0100107.2939120.45250.662291.69711.20880.0136155.6330130.65250.6622111.42320.76710.0102105.6065140.5522.50.542098.00460.96200.0110130.6791150.5527.50.542085.37611.35840.0155158.0305160.5522.50.7820123.71600.66660.0106134.8080170.5527.50.782096.11521.11310.0157159.7465180.45250.542088.26321.26650.0129152.1337190.65250.5420101.89410.88190.0102109.0929200.45250.7820102.23311.00510.0130150.8916210.65250.7820128.56770.61220.0100105.7656220.5522.50.6618121.71630.68030.0111135.3569230.5527.50.661894.88921.14490.0156161.3298240.5522.50.6622106.92040.83580.0106134.3837250.5527.50.662287.82791.30950.0161150.2442Table 8
Proposed fuzzy if-then rules.
TrialD1 of f1 (x)D2 of f2 (x)Output1SSSSS2ASSRS3LSSS4SRSSS5ARSS6LRSSA7SSS8ASSA9LSAL10SSAS11ASASA12LSAA13SASA14AAA15LAAL16SALSA17AALA18LALAL19SNLAL20ANLAL21LNLL22SLAL23ALL24LLNL25SSLAL26ASLL27LSLSLD1 and D2 are desirabilities of frequency and displacement, respectivelyRelationship plot of the output versus inputs in the FIS modeling is illustrated in Figure8. Figure 9 gives the fuzzy if-then rules.Figure 8
Relationship plot in the FIS modeling.Figure 9
Illustration of the fuzzy if-then rules in the FIS modeling.Figure9 illustrates the fuzzy rules. When D1 value is input into the left column and D2 value is input into the middle column, the output of FIS is found in the right column of Figure 9, respectively. Table 9 gives the results of the output of FIS.Table 9
Case study 1: output of FIS.
TrialD1 of f1 (x)D2 of f2 (x)Output10.5491537090.2798440080.37520.5456311810.2718248540.37530.941877450.0690562520.31940.0969439740.8502883160.62550.5850437020.2492648850.37560.4729160270.3115605120.37570.7584448450.1565175230.39880.2277621560.4938219180.42590.6188914480.2230176360.352100.4425323110.4138034180.412110.9226981480.0326832980.279120.2881715150.5589600740.459130.6923516750.136288160.281140.4388306920.2976351290.375150.173718820.6570969790.45160.9148083530.0648372960.315170.374399270.464047540.466180.2211588410.6136309150.422190.5465391280.2150518280.343200.4426951860.4040242090.421100.25220.8419759760.0984019880.335230.3681060250.4741340930.478240.5785128680.2191788050.348250.2468618640.6021187950.436Next, ANFIS modeling is built based on Tables7 and 9 in MATLAB R2019b. In the developed model, there are nodes of 193, linear parameters of 405, nonlinear parameters of 36, total parameters of 441, training data of 17, testing data of 8, and fuzzy if-then rules of 81. The built ANFIS model has relatively good performance indexes with R2 of 0.963 and RMSE of 0.035.In this study, two objective functions are combined into a single objective function by using the FIS. The output of the FIS is the single objective function, which can be optimized by many methods. Then, the displacement and frequency can be predicted based on the optimal design variables. Specifically, Taguchi can optimize the single fitness function but may trap a local optimum value; meanwhile MFO algorithm can search a global optimum value. Hence, the optimal values from the Taguchi are compared with those found by the proposed method’s framework. The purpose of this comparison is to clearly demonstrate the usefulness of the devoted method in searching the global solution.In Table10, the optimal parameters by the TF include T2 = 0.45 mm, L2 = 27.5 mm, T3 = 0.54 mm, and L3 = 20 mm. The results of TF show the frequency, displacement, parasitic error, and stress are approximately 75.871 Hz and 1.776 mm, 0.019 mm, and 212.114 MPa, respectively. However, those optimal values can trap local optimum solutions. To find a global optimum solution, moth-flame optimization algorithm is adopted. The results of the hybrid computational method determine T2 = 0.45 mm, L2 = 27.5 mm, T3 = 0.78 mm, and L3 = 18 mm (Table 10). The frequency, displacement, parasitic error, and stress are about 85.174 Hz and 2.447 mm, 0.016 mm, and 145.982 MPa, respectively. Furthermore, the MPCI in the hybrid computational method is better than that in TF. It means that the hybrid computational method outperformed the TF.Table 10
Case study 1: the optimal results.
MethodOptimal parametersOptimal resultsMPCIf1 (x) (HZ)f2 (x) (mm)f3 (x) (mm)f4 (x) (MPa)n-safety factorTFT2 = 0.450.62875.87111.776360.0198212.1142.731L2 = 27.5T3 = 0.54L3 = 20The current methodT2 = 0.450.84885.17492.44720.0160145.9823.468L2 = 27.5T3 = 07.8L3 = 18


#### 4.2.3. Investigation on Case Study 2

Initialized from Table5 and Figure 7, the space of design parameters is also limited to generate a new population for the modeling and optimization procedure. The optimization formulation is stated as follows.Searchx = [T2, L2, T3,]T.(8)Maximizefxcase2,(9)s.t.f1x≥1.7Hz,f2x≥1.7mm,f3x≤0.02mm,f4x≤503MPan,0.45mm≤T2≤0.65mm,22.5mm≤L2≤27.5mm,0.54mm≤T3≤0.78mm.Table11 gives the results of 13 numerical experiments, including the frequency, the displacement, the parasitic motion, and stress. The desirabilities are calculated (Table 12).Table 11
Case study 2: simulated results.
TrialRefined variables (unit: mm)FrequencyDisplacementParasiticStressT2L2T3f1 (x) (Hz)f2 (x) (mm)f3 (x) (mm)f4 (x) (MPa)10.55250.66104.04820.93360.0124162.735520.4522.50.6699.318411.04590.0155153.047230.6522.50.66127.18880.58060.008192.9515840.4527.50.6674.182281.84190.0211227.180350.6527.50.66107.04810.87420.0120117.824960.45250.5488.263161.26650.0129152.133770.65250.54101.89410.88190.0102109.092980.45250.78102.23311.00510.0130150.891690.65250.78128.56770.61220.0100105.7656100.5522.50.5498.004640.96200.0110130.6791110.5527.50.5485.376071.35840.0155158.0305120.5522.50.78123.7160.66660.0106134.808130.5527.50.7896.115231.11310.0157159.7465Table 12
Case study 2: results of the output of FIS system.
TrialD1 of f1 (x)D2 of f2 (x)Output10.5491537090.2798440080.37520.4983239310.3284312420.37530.9153048810.0894116220.34240.0593410720.9105883780.62550.568175480.2733139290.37560.2208068270.609727790.42170.5669217960.1748976850.29980.4584019210.4006139180.3959100.25100.4399921780.2769963760.375110.1845846530.6401519010.409120.9320285880.0446913770.294130.4013238530.4475952950.448Subsequently, the fuzzy if-then rules are built based on Table8, and the output of FIS system is given in Table 12.In this ANFIS model, there are nodes of 34, linear parameters of 32, nonlinear parameters of 18, total parameters of 50, training data of 9, testing data of 4, and fuzzy if-then rules of 8. The developed ANFIS modeling achieves relatively good performance indexes withR2 of 0.954 and RMSE of 0.012.Using TF, the optimal parameters areT2 = 0.45 mm, L2 = 27.5 mm, and T3 = 0.66 mm (Table 13). The frequency, displacement, parasitic error, and stress are about 79.460 Hz, 1.637 mm, 0.018 mm, and 198.015 MPa, respectively. The safety factor is determined to be about 2.54. The optimal solutions are local solutions.Table 13
Case study 2: the optimal results.
MethodOptimal parameters (mm)Optimal resultsMPCIFrequencyf1 (x) (HZ)Displacementf2 (x) (mm)Parasiticf3 (x) (mm)Stressf4 (x) (MPa)n-safety factorTFT2 = 0.450.556879.46011.63710.0184198.0152.540L2 = 27.5T3 = 0.66Computational methodT2 = 0.450.640276.74351.70020.0197236.0272.131L2 = 27.5T3 = 0.69In Table13, using the hybrid computational method, the optimal parameters are T2 = 0.45 mm, L2 = 27.5 mm, and T3 = 0.69 mm. The optimal frequency, displacement, parasitic error, equivalent stress, and safety factor are 76.743 Hz, 1.700 mm, 0.019 mm, 236.027 MPa, and 2.131, respectively. It is noted that the MPCI in the hybrid computational method is also better than that in the TF.


### 4.3. Discussion

In the previous sections, the sensitivity of factors on the output responses is analyzed to redetermine the main geometrical parameters. Those parameters contribute directly the frequency, displacement, parasitic motion, and stress of 1-DOF mechanism. Two numerical examples are studied. In order to calculate an error among the predicted value (Rp) and simulation (Rs), the error (E) is calculated as(10)E%=Rp−RsRs−1∗100.A comparison with case 1 is performed. Using the computational method, the error is around 7% for case study 1 and 3.8% for case 2. On the contrary, using TF, the error is 23% for case 1 and 14.7% for case 2 (Table14). Additionally, the computational methodology can reach a global solution. It can infer that the proposed computational scheme is greater than TF. Besides, case 1 is adopted as an optimal candidate thanks to its highest MPCI value of 0.848.Table 14
Comparison results of two cases.
Case studyParameters (mm)Optimal responsesf1 (x) (Hz)f2 (x) (mm)f3 (x) (mm)f4 (x) (MPa)nMPCICase 1The current methodT2 = 0.450.848L2 = 27.5Rp = 79.5179Rp = 1.8977Rp = 0.0224Rp = 210.382Rp = 2.390T3 = 0.78Rs = 76.062Rs = 1.7611Rs = 0.0216Rs = 223.38Rs = 2.155L3 = 18E = 4.54%E = 7.75%E = 3.70%E = 5.81%E = 1.09%Case 2The current methodT2 = 0.45Rp = 76.7435Rp = 1.8266Rp = 0.0211Rp = 236.027Rp = 2.1310.640L2 = 27.5Rs = 74.205Rs = 1.8350Rs = 0.0214Rs = 227.210Rs = 2.213T3 = 0.69E = 3.42%E = 0.45%E = 1.40%E = 3.88%E = 3.70%Figures10(a) and 10(b) illustrate the deformation and stress distributions of the mechanism, respectively.Figure 10
Simulations for case 1: (a) deformation and (b) stress distribution.
(a)(b)


### 4.4. Comparisons among Different Methods

The previous section shows that the computational method outperforms the TF in searching a global optimal solution of the 1-DOF mechanism. The comparison of the behaviors of the suggested method with other algorithms includes ANFIS-teaching-learning-based algorithm (ANFIS-coupled TLBO) [44] and ANFIS-Jaya [45]. Nonparameter statistical investigations are performed and involved in resolving case 1. In Table 15, the MPCI is almost the same for three methods. However, the output responses of the mechanism from the suggested method are better than those from the other algorithms.Table 15
Comparison among different methods.
Approachesf1 (x) (Hz)f2 (x) (mm)MPCIANFIS-Jaya (T2 = 0.45 mm, L2 = 27.49 mm, T3 = 0.77 mm, L3 = 18 mm)76.8801.7190.848ANFIS-TLBO (T2 = 0.45 mm, L2 = 27.45 mm, T3 = 0.78 mm, L3 = 18 mm)76.3761.7460.848The current method (T2 = 0.45 mm, L2 = 27.5 mm, T3 = 0.78 mm, L3 = 18 mm)79.5171.8970.848Two nonparameter statistical techniques [57] are employed to determine the behaviors of three methods. Each method has 60 runs. As given in Table 16, it is inferred that the suggested computational method is superior to other methods.Table 16
Results of Wilcoxon tests.
FrequencyThe current method and ANFIS-coupled JayaWilcoxon indexp valueDifference0≤0.001−2.6379The current method and ANFIS-coupled TLBOWilcoxon indexp valueDifference0≤0.001−3.1419DisplacementThe current method and ANFIS-coupled JayaWilcoxon indexp valueDifference0≤0.001−0.1786The current method versus ANFIS-coupled TLBOWilcoxon indexp valueDifference0≤0.001−0.1513In Table17, the results of Friedman tests also prove that the computational method outperformed the others.Table 17
Results of Friedman tests.
FrequencyMethodAverageRankANFIS-coupled Jaya76.880060ANFIS-coupled TLBO76.3760180The current method79.5179120Overall77.5913p value≤0.001DisplacementMethodAverageRankANFIS-coupled Jaya1.7191060ANFIS-coupled TLBO1.74640180The current method1.89770120Overall1.78773p value≤0.001In order to compare the convergence speed among different algorithms in the literature, the proposed method (a hybridization of desirability, fuzzy, and ANFIS-based MFO) is compared with the ant lion optimizer (ALO) [48], particle swarm optimization-based gravitational search algorithm (PSOGSA) [49], and sine-cosine algorithm (SCA) [50]. A maximum iteration of 10000 and initial population of 20 are used for all algorithms. The results found that the current method in this study has faster convergence than others because the devoted method only needs a computation time of 559.78 s, as given in Table 18.Table 18
Statistical results for comparison among metaheuristic methods.
MethodResultsTime (s)Frequency (Hz)Displacement (mm)ANFIS-coupled PSOGSA79.4211.892616.038ANFIS-coupled SCA79.4261.895599.544ANFIS-coupled ALO79.4251.893623.20ANFIS-coupled LAPO79.5171.8972707.75The current method79.5171.897559.78Additionally, the superiority of the current method in this paper is also compared with the Taguchi, the desirability, and Taguchi-fuzzy methods. It is remarked that the Taguchi method is able to search the optimal solution for each response (single optimization problem), while the others are used to solve the MOO problems for case study 1. Datasets in Table7 are used for the Taguchi, Taguchi-fuzzy, and desirability methods. The results are summarized in Table 19.Table 19
Comparison among the current method, Taguchi method, and Taguchi-fuzzy method.
MethodOptimal resultsTaguchi (single optimization)Parameters (mm)T2 = 0.65T2 = 0.45T2 = 0.65T2 = 0.65L2 = 22.5L2 = 27.5L2 = 22.5L2 = 22.5T3 = 0.78T3 = 0.54T3 = 0.78T3 = 0.66L3 = 18L3 = 22L3 = 18L3 = 20Responsef1 (x) (Hz)f2 (x) (mm)f3 (x) (mm)f4 (x) (MPa)Value139.4741.7600.00890.405Taguchi-fuzzy (MOO)Parameters (mm)T2 = 0.45, L2 = 27.5, T3 = 0.54, L3 = 20Responsef1 (x) (Hz)f2 (x) (mm)f3 (x) (mm)f4 (x) (MPa)Value75.8711.7760.019212.114Desirability (MOO)Parameters (mm)T2 = 0.65, L2 = 22.5, T3 = 0.54, L3 = 21.27Responsef1 (x) (Hz)f2 (x) (mm)f3 (x) (mm)f4 (x) (MPa)Value106.8410.9300.011120.090The current method (MOO)Parameters (mm)T2 = 0.45, L2 = 27.5, T3 = 0.78, L3 = 18Responsef1 (x) (Hz)f2 (x) (mm)f3 (x) (mm)f4 (x) (MPa)Value79.5171.8970.022210.382From the results of Table19, it can be revealed that, for optimization of a single response, the Taguchi method is a more suitable tool. By using the Taguchi, the optimal parameters are found with respect to each response but it only finds the optimum value for a single cost function, as shown in Table 19. By using the Taguchi-fuzzy method, the results searched the optimal sets of parameters for MOO but this approach is based on the Taguchi reasoning. This means that the optimal solutions can be also local values. Besides, using the desirability, the results found the optimal parameters for the MOO but this technique is based on the prediction precision of the mathematical models. Lastly, the current method (a hybridization of desirability, fuzzy, and ANFIS-based MFO methods) is a reliable technique that is superior to others because it can search the global solution for the 1-DOF mechanism.Strengths of the proposed method’s framework can be listed as follows:(i)
The behaviors of the 1-DOF mechanism are easily analyzed by linear/nonlinear FEM(ii)
Influence of the units of performances on the finally optimal solution can be ignored via the desirability method(iii)
The multiple design targets are easily converted into a single cost function through the FIS(iv)
The unknown relation among the design variables and the single cost function can be precisely approximated by ANFIS(v)
The global optimum solution of the 1-DOF mechanism can be achieved via the ANFIS-based MFOHowever, this study still has drawbacks, including the computational principle and adaptive process. The computational principle requires a variety of different methods from statistics, numerical method, intelligent method, and metaheuristics. Besides, the adaptive process needs an adaptive update of new design variables and performances have not been studied yet. Finally, the computational methods have not been automatically connected.At last, limitations of the present design framework concentrate on time and computational complexity as well as efficiency. This method needs a long period of computing time for each method. The computational complexity is mainly dependent on computer ability and experiences in the field. For a complex mechanism, the FEM and working time of computer are restricted.


## 4.1. Determination of Main Parameters

Involving whole initial design variables, design of numerical experiments is built by BBD technique. Table2 gives initial design parameters and their ranges. Each experiment is implemented by simulations. The initial results are retrieved in Table 3.Table 2
Initial design variables and range (unit: mm).
NotationRangeLevel 1Level 2Level 3T10.72≤T1≤1.040.720.881.04L114.5≤L1≤16.514.515.516.5T20.45≤T2≤0.650.450.550.65L222.5≤L2≤27.522.524.527.5T30.54≤T3≤0.780.540.660.78L318≤L3≤22182022Table 3
Initial results.
TrialParameters (unit: mm)Frequency (Hz)Displacement (mm)Parasitic motion (mm)Stress (MPa)T1L1T2L2T3L3f1 (x)f2 (x)f3 (x)f4 (x)10.88150.55250.6620103.99160.93350.0124162.7020.7213.50.5522.50.6620112.93610.77610.0110131.6931.0413.50.5522.50.6620112.53400.77870.0112134.3840.7216.50.5522.50.6620113.42980.76630.0106130.9851.0416.50.5522.50.6620112.36020.77830.0111134.1260.7213.50.5527.50.662091.03061.23350.0158151.3071.0413.50.5527.50.662090.57611.24750.0165167.1780.7216.50.5527.50.662092.12461.20130.0157159.1791.0416.50.5527.50.662091.88951.19850.0157159.29100.8813.50.45250.542087.74641.28450.0130152.76110.8816.50.45250.542085.41901.36010.0139151.35120.8813.50.65250.5420101.81920.87860.0102108.25130.8816.50.65250.5420101.15710.89220.0103106.90140.8813.50.45250.7820102.56910.99770.0128151.42150.8816.50.45250.7820100.47351.03620.0130154.94160.8813.50.65250.7820127.65350.62160.0104107.23170.8816.50.65250.7820129.24420.60420.0101106.92180.88150.4522.50.6618103.20410.98340.0155152.76190.88150.6522.50.6618136.82870.51970.008293.28200.88150.4527.50.661876.05151.76370.0210225.27210.88150.6527.50.6618112.59880.79960.0117109.54220.88150.4522.50.662294.85561.12170.0155152.93230.88150.6522.50.6622117.40830.65270.008087.21240.88150.4527.50.662272.71041.91200.0210227.36250.88150.6527.50.6622102.51060.93220.0119109.20260.72150.5522.50.542098.05390.96190.0110130.67271.04150.5522.50.542098.07510.95510.0109132.22280.72150.5527.50.542085.15011.36710.0153157.48291.04150.5527.50.542085.20191.35840.0156157.87300.72150.5522.50.7820123.76920.66680.0106134.85311.04150.5522.50.7820122.53880.67660.0106136.39320.72150.5527.50.782096.08111.11880.0157160.43331.04150.5527.50.782094.78341.14950.0161170.74340.8813.50.55250.541898.06781.03400.0130168.07350.8816.50.55250.541897.27161.04840.0128155.91360.8813.50.55250.7818112.32990.82370.0129166.04370.8816.50.55250.7818103.99160.82180.0127164.79380.8813.50.55250.5422112.93611.27570.0127165.29390.8816.50.55250.5422112.53401.27790.0139150.35400.8813.50.55250.7822113.42980.91780.0135159.05410.8816.50.55250.7822112.36020.91980.0130166.97420.72150.45250.661891.03061.04120.0130152.59431.04150.45250.661890.57611.08720.0133149.97440.72150.65250.661892.12460.63970.0102106.64451.04150.65250.661891.88950.64040.0102106.75460.72150.45250.662287.74641.20740.0135155.46471.04150.45250.662285.41901.21520.0134151.53480.72150.65250.6622101.81920.77500.0102107.25491.04150.65250.6622111.36040.76150.0100106.38Case study 1 focuses on the sensitivity analysis for the natural frequency. In Table4, the results of ANOVA determined that T1 with contribution of 0.02% and L1 with contribution of 0% and their p values are 0.567 and 0.891, which are larger than 0.05. The contributions of T1 and L1 are very small, and they are therefore deleted from modeling and optimization process. Additionally, a matrix plot is drawn to show effects of all parameters on the frequency. As seen in Figure 4, it also has a similar conclusion in Table 4. To summarize, case study 1 deals with the main design parameters, including T2, L2, T3, and L3.Table 4
The frequency’s ANOVA.
SourceDfSeq SSContributionAdj SSAdj MSF valuep valueModel279662.8898.639662.88357.8855.89≤0.001Linear69110.0892.999110.081518.35237.10≤0.001T112.160.022.162.160.340.567L110.120.000.120.120.020.891T213680.1437.563680.143680.14574.67≤0.001L212715.4427.722715.442715.44424.03≤0.001T312085.9121.292085.912085.91325.72≤0.001L31626.316.39626.31626.3197.80≤0.001Square6250.432.56250.4341.746.520.001T1 ∗ T117.110.0712.2612.261.910.181L1 ∗ L114.490.050.400.400.060.804T2 ∗ T21186.441.9039.1739.176.120.022L2 ∗ L2130.950.3251.6051.608.060.010T3 ∗ T318.500.0919.6419.643.070.095L3 ∗ L3112.950.1312.9512.952.020.1702-Way interaction15302.363.09302.3620.163.150.008T1 ∗ L110.030.000.030.030.000.951T1 ∗ T211.290.011.291.290.200.658T1 ∗ L210.030.000.030.030.010.942T1 ∗ T310.850.010.850.850.130.720T1 ∗ L311.070.011.071.070.170.687L1 ∗ T213.580.043.583.580.560.463L1 ∗ L210.540.010.540.540.090.773L1 ∗ T310.650.010.650.650.100.754L1 ∗ L310.030.000.030.030.000.950T2 ∗ L2112.930.1312.9312.932.020.170T2 ∗ T3172.270.7472.2772.2711.280.003T2 ∗ L3159.820.6159.8259.829.340.006L2 ∗ T31110.011.12110.01110.0117.18≤0.001L2 ∗ L3125.700.2625.7025.704.010.058T3 ∗ L3113.580.1413.5813.582.120.160Error21134.481.37134.486.40Total489797.36100.00Figure 4
Sensitivity plot of the frequency.Case study 2 deals with the sensitivity analysis for displacement. Based on the ANOVA results in Table5, the parameter’s contributions T1, L1, and L3 are very small with 1.57%, 1.33%, and 1.86%, respectively.Table 5
The displacement’s ANOVA.
SourceDfSeq SSContribution (%)Adj SSAdj MSF valuep valueModel275.8298883.265.829880.215923.870.001Linear64.5944265.614.594420.7657413.72≤0.001T110.109931.570.109930.109931.970.175L110.093391.330.093390.093391.670.210T211.6499523.561.649951.6499529.56≤0.001L212.1464430.652.146442.1464438.45≤0.001T310.464726.640.464720.464728.330.009L310.129991.860.129990.129992.330.142Square60.390325.570.390320.065051.170.361T1 ∗ T110.115081.640.045570.045570.820.376L1 ∗ L110.171382.450.059440.059441.060.314T2 ∗ T210.028330.400.000820.000820.010.905L2 ∗ L210.000160.000.027040.027040.480.494T3 ∗ T310.051030.730.075370.075371.350.258L3 ∗ L310.024350.350.024350.024350.440.5162-Way interaction150.8451412.070.845140.056341.010.482T1 ∗ L110.290774.150.290770.290775.210.033T1 ∗ T210.000550.010.000550.000550.010.922T1 ∗ L210.143842.050.143840.143842.580.123T1 ∗ T310.000390.010.000390.000390.010.934T1 ∗ L310.000340.000.000340.000340.010.938L1 ∗ T210.001730.020.001730.001730.030.862L1 ∗ L210.267073.810.267070.267074.780.040L1 ∗ T310.000450.010.000450.000450.010.929L1 ∗ L310.000010.000.000010.000010.000.990T2 ∗ L210.127781.820.127780.127782.290.145T2 ∗ T310.000540.010.000540.000540.010.923T2 ∗ L310.000220.000.000220.000220.000.951L2 ∗ T310.001700.020.001700.001700.030.863L2 ∗ L310.000010.000.000010.000010.000.989T3 ∗ L310.009740.140.009740.009740.170.680Error211.1722116.741.172210.05582Total487.00209100.00The results of Figure5 also have the same conclusion in Table 5. It means that the parameters T1, L1, and L3 should be suppressed during modeling and optimization procedure.Figure 5
Sensitivity plot of the displacement.


## 4.2. Modeling and Optimization

### 4.2.1. Determination of MFs Types

Table6 assigns the fuzzy variables for MFs. In this article, the frequency and displacement desirabilities are two inputs of the FIS. The MFs types for two inputs and an output of the FIS system are illustrated in Figures 6 and 7.Table 6
Proposed fuzzy variables.
AbbreviationSSRSSSAAALNLLSLFuzzy variableSo smallRelatively smallSmallSmall averageAverageAverage largeNear largeLargeSo largeFigure 6
MFs types: (a) the frequency and (b) the displacement.
(a)(b)Figure 7
MFs type for the output of FIS system.


### 4.2.2. Investigation on Case Study 1

As discussed in the previous section, overall initial design variables are limited. Those factors actually contribute the responses of 1-DOF mechanism. Besides, spaces of parameters are newly initialized for the modeling and optimization process. The optimization formulation is stated as follows.Findx = [T2, L2, T3, L3]T.(6)Maximizefxcase1,(7)s.t.f1x≥70Hz,f2x≥1.7mm,f3x≤0.02mm,f4x≤503MPan,0.45mm≤T2≤0.65mm,22.5mm≤L2≤27.5mm,0.54mm≤T3≤0.78mm,18mm≤L3≤22mm.f (x) is a single fitness function. Table 7 shows the results for case study 1. Table 8 presents the MFs variables for the fuzzy if-then rules in modeling both objective functions into a single objective function.Table 7
Case study 1: simulated results.
TrialRefined parameters (unit: mm)FrequencyDisplacementParasitic motionStressT2L2T3L3f1 (x) (Hz)f2 (x) (mm)f3 (x) (mm)f4 (x) (MPa)10.55250.6620104.04820.93360.0124162.735520.4522.50.662099.31841.04590.0155153.047230.6522.50.6620127.18880.58060.008192.951640.4527.50.662074.18231.84190.0211227.180350.6527.50.6620107.04810.87420.0120117.824960.55250.541898.74761.01890.0127165.136970.55250.7818112.52950.82210.0127164.978580.55250.542285.98011.26110.0132158.412590.55250.7822105.50510.91830.0129164.3392100.45250.6618100.18991.04110.0130152.5881110.65250.6618124.04860.65190.0100107.2939120.45250.662291.69711.20880.0136155.6330130.65250.6622111.42320.76710.0102105.6065140.5522.50.542098.00460.96200.0110130.6791150.5527.50.542085.37611.35840.0155158.0305160.5522.50.7820123.71600.66660.0106134.8080170.5527.50.782096.11521.11310.0157159.7465180.45250.542088.26321.26650.0129152.1337190.65250.5420101.89410.88190.0102109.0929200.45250.7820102.23311.00510.0130150.8916210.65250.7820128.56770.61220.0100105.7656220.5522.50.6618121.71630.68030.0111135.3569230.5527.50.661894.88921.14490.0156161.3298240.5522.50.6622106.92040.83580.0106134.3837250.5527.50.662287.82791.30950.0161150.2442Table 8
Proposed fuzzy if-then rules.
TrialD1 of f1 (x)D2 of f2 (x)Output1SSSSS2ASSRS3LSSS4SRSSS5ARSS6LRSSA7SSS8ASSA9LSAL10SSAS11ASASA12LSAA13SASA14AAA15LAAL16SALSA17AALA18LALAL19SNLAL20ANLAL21LNLL22SLAL23ALL24LLNL25SSLAL26ASLL27LSLSLD1 and D2 are desirabilities of frequency and displacement, respectivelyRelationship plot of the output versus inputs in the FIS modeling is illustrated in Figure8. Figure 9 gives the fuzzy if-then rules.Figure 8
Relationship plot in the FIS modeling.Figure 9
Illustration of the fuzzy if-then rules in the FIS modeling.Figure9 illustrates the fuzzy rules. When D1 value is input into the left column and D2 value is input into the middle column, the output of FIS is found in the right column of Figure 9, respectively. Table 9 gives the results of the output of FIS.Table 9
Case study 1: output of FIS.
TrialD1 of f1 (x)D2 of f2 (x)Output10.5491537090.2798440080.37520.5456311810.2718248540.37530.941877450.0690562520.31940.0969439740.8502883160.62550.5850437020.2492648850.37560.4729160270.3115605120.37570.7584448450.1565175230.39880.2277621560.4938219180.42590.6188914480.2230176360.352100.4425323110.4138034180.412110.9226981480.0326832980.279120.2881715150.5589600740.459130.6923516750.136288160.281140.4388306920.2976351290.375150.173718820.6570969790.45160.9148083530.0648372960.315170.374399270.464047540.466180.2211588410.6136309150.422190.5465391280.2150518280.343200.4426951860.4040242090.421100.25220.8419759760.0984019880.335230.3681060250.4741340930.478240.5785128680.2191788050.348250.2468618640.6021187950.436Next, ANFIS modeling is built based on Tables7 and 9 in MATLAB R2019b. In the developed model, there are nodes of 193, linear parameters of 405, nonlinear parameters of 36, total parameters of 441, training data of 17, testing data of 8, and fuzzy if-then rules of 81. The built ANFIS model has relatively good performance indexes with R2 of 0.963 and RMSE of 0.035.In this study, two objective functions are combined into a single objective function by using the FIS. The output of the FIS is the single objective function, which can be optimized by many methods. Then, the displacement and frequency can be predicted based on the optimal design variables. Specifically, Taguchi can optimize the single fitness function but may trap a local optimum value; meanwhile MFO algorithm can search a global optimum value. Hence, the optimal values from the Taguchi are compared with those found by the proposed method’s framework. The purpose of this comparison is to clearly demonstrate the usefulness of the devoted method in searching the global solution.In Table10, the optimal parameters by the TF include T2 = 0.45 mm, L2 = 27.5 mm, T3 = 0.54 mm, and L3 = 20 mm. The results of TF show the frequency, displacement, parasitic error, and stress are approximately 75.871 Hz and 1.776 mm, 0.019 mm, and 212.114 MPa, respectively. However, those optimal values can trap local optimum solutions. To find a global optimum solution, moth-flame optimization algorithm is adopted. The results of the hybrid computational method determine T2 = 0.45 mm, L2 = 27.5 mm, T3 = 0.78 mm, and L3 = 18 mm (Table 10). The frequency, displacement, parasitic error, and stress are about 85.174 Hz and 2.447 mm, 0.016 mm, and 145.982 MPa, respectively. Furthermore, the MPCI in the hybrid computational method is better than that in TF. It means that the hybrid computational method outperformed the TF.Table 10
Case study 1: the optimal results.
MethodOptimal parametersOptimal resultsMPCIf1 (x) (HZ)f2 (x) (mm)f3 (x) (mm)f4 (x) (MPa)n-safety factorTFT2 = 0.450.62875.87111.776360.0198212.1142.731L2 = 27.5T3 = 0.54L3 = 20The current methodT2 = 0.450.84885.17492.44720.0160145.9823.468L2 = 27.5T3 = 07.8L3 = 18


### 4.2.3. Investigation on Case Study 2

Initialized from Table5 and Figure 7, the space of design parameters is also limited to generate a new population for the modeling and optimization procedure. The optimization formulation is stated as follows.Searchx = [T2, L2, T3,]T.(8)Maximizefxcase2,(9)s.t.f1x≥1.7Hz,f2x≥1.7mm,f3x≤0.02mm,f4x≤503MPan,0.45mm≤T2≤0.65mm,22.5mm≤L2≤27.5mm,0.54mm≤T3≤0.78mm.Table11 gives the results of 13 numerical experiments, including the frequency, the displacement, the parasitic motion, and stress. The desirabilities are calculated (Table 12).Table 11
Case study 2: simulated results.
TrialRefined variables (unit: mm)FrequencyDisplacementParasiticStressT2L2T3f1 (x) (Hz)f2 (x) (mm)f3 (x) (mm)f4 (x) (MPa)10.55250.66104.04820.93360.0124162.735520.4522.50.6699.318411.04590.0155153.047230.6522.50.66127.18880.58060.008192.9515840.4527.50.6674.182281.84190.0211227.180350.6527.50.66107.04810.87420.0120117.824960.45250.5488.263161.26650.0129152.133770.65250.54101.89410.88190.0102109.092980.45250.78102.23311.00510.0130150.891690.65250.78128.56770.61220.0100105.7656100.5522.50.5498.004640.96200.0110130.6791110.5527.50.5485.376071.35840.0155158.0305120.5522.50.78123.7160.66660.0106134.808130.5527.50.7896.115231.11310.0157159.7465Table 12
Case study 2: results of the output of FIS system.
TrialD1 of f1 (x)D2 of f2 (x)Output10.5491537090.2798440080.37520.4983239310.3284312420.37530.9153048810.0894116220.34240.0593410720.9105883780.62550.568175480.2733139290.37560.2208068270.609727790.42170.5669217960.1748976850.29980.4584019210.4006139180.3959100.25100.4399921780.2769963760.375110.1845846530.6401519010.409120.9320285880.0446913770.294130.4013238530.4475952950.448Subsequently, the fuzzy if-then rules are built based on Table8, and the output of FIS system is given in Table 12.In this ANFIS model, there are nodes of 34, linear parameters of 32, nonlinear parameters of 18, total parameters of 50, training data of 9, testing data of 4, and fuzzy if-then rules of 8. The developed ANFIS modeling achieves relatively good performance indexes withR2 of 0.954 and RMSE of 0.012.Using TF, the optimal parameters areT2 = 0.45 mm, L2 = 27.5 mm, and T3 = 0.66 mm (Table 13). The frequency, displacement, parasitic error, and stress are about 79.460 Hz, 1.637 mm, 0.018 mm, and 198.015 MPa, respectively. The safety factor is determined to be about 2.54. The optimal solutions are local solutions.Table 13
Case study 2: the optimal results.
MethodOptimal parameters (mm)Optimal resultsMPCIFrequencyf1 (x) (HZ)Displacementf2 (x) (mm)Parasiticf3 (x) (mm)Stressf4 (x) (MPa)n-safety factorTFT2 = 0.450.556879.46011.63710.0184198.0152.540L2 = 27.5T3 = 0.66Computational methodT2 = 0.450.640276.74351.70020.0197236.0272.131L2 = 27.5T3 = 0.69In Table13, using the hybrid computational method, the optimal parameters are T2 = 0.45 mm, L2 = 27.5 mm, and T3 = 0.69 mm. The optimal frequency, displacement, parasitic error, equivalent stress, and safety factor are 76.743 Hz, 1.700 mm, 0.019 mm, 236.027 MPa, and 2.131, respectively. It is noted that the MPCI in the hybrid computational method is also better than that in the TF.


## 4.2.1. Determination of MFs Types

Table6 assigns the fuzzy variables for MFs. In this article, the frequency and displacement desirabilities are two inputs of the FIS. The MFs types for two inputs and an output of the FIS system are illustrated in Figures 6 and 7.Table 6
Proposed fuzzy variables.
AbbreviationSSRSSSAAALNLLSLFuzzy variableSo smallRelatively smallSmallSmall averageAverageAverage largeNear largeLargeSo largeFigure 6
MFs types: (a) the frequency and (b) the displacement.
(a)(b)Figure 7
MFs type for the output of FIS system.


## 4.2.2. Investigation on Case Study 1

As discussed in the previous section, overall initial design variables are limited. Those factors actually contribute the responses of 1-DOF mechanism. Besides, spaces of parameters are newly initialized for the modeling and optimization process. The optimization formulation is stated as follows.Findx = [T2, L2, T3, L3]T.(6)Maximizefxcase1,(7)s.t.f1x≥70Hz,f2x≥1.7mm,f3x≤0.02mm,f4x≤503MPan,0.45mm≤T2≤0.65mm,22.5mm≤L2≤27.5mm,0.54mm≤T3≤0.78mm,18mm≤L3≤22mm.f (x) is a single fitness function. Table 7 shows the results for case study 1. Table 8 presents the MFs variables for the fuzzy if-then rules in modeling both objective functions into a single objective function.Table 7
Case study 1: simulated results.
TrialRefined parameters (unit: mm)FrequencyDisplacementParasitic motionStressT2L2T3L3f1 (x) (Hz)f2 (x) (mm)f3 (x) (mm)f4 (x) (MPa)10.55250.6620104.04820.93360.0124162.735520.4522.50.662099.31841.04590.0155153.047230.6522.50.6620127.18880.58060.008192.951640.4527.50.662074.18231.84190.0211227.180350.6527.50.6620107.04810.87420.0120117.824960.55250.541898.74761.01890.0127165.136970.55250.7818112.52950.82210.0127164.978580.55250.542285.98011.26110.0132158.412590.55250.7822105.50510.91830.0129164.3392100.45250.6618100.18991.04110.0130152.5881110.65250.6618124.04860.65190.0100107.2939120.45250.662291.69711.20880.0136155.6330130.65250.6622111.42320.76710.0102105.6065140.5522.50.542098.00460.96200.0110130.6791150.5527.50.542085.37611.35840.0155158.0305160.5522.50.7820123.71600.66660.0106134.8080170.5527.50.782096.11521.11310.0157159.7465180.45250.542088.26321.26650.0129152.1337190.65250.5420101.89410.88190.0102109.0929200.45250.7820102.23311.00510.0130150.8916210.65250.7820128.56770.61220.0100105.7656220.5522.50.6618121.71630.68030.0111135.3569230.5527.50.661894.88921.14490.0156161.3298240.5522.50.6622106.92040.83580.0106134.3837250.5527.50.662287.82791.30950.0161150.2442Table 8
Proposed fuzzy if-then rules.
TrialD1 of f1 (x)D2 of f2 (x)Output1SSSSS2ASSRS3LSSS4SRSSS5ARSS6LRSSA7SSS8ASSA9LSAL10SSAS11ASASA12LSAA13SASA14AAA15LAAL16SALSA17AALA18LALAL19SNLAL20ANLAL21LNLL22SLAL23ALL24LLNL25SSLAL26ASLL27LSLSLD1 and D2 are desirabilities of frequency and displacement, respectivelyRelationship plot of the output versus inputs in the FIS modeling is illustrated in Figure8. Figure 9 gives the fuzzy if-then rules.Figure 8
Relationship plot in the FIS modeling.Figure 9
Illustration of the fuzzy if-then rules in the FIS modeling.Figure9 illustrates the fuzzy rules. When D1 value is input into the left column and D2 value is input into the middle column, the output of FIS is found in the right column of Figure 9, respectively. Table 9 gives the results of the output of FIS.Table 9
Case study 1: output of FIS.
TrialD1 of f1 (x)D2 of f2 (x)Output10.5491537090.2798440080.37520.5456311810.2718248540.37530.941877450.0690562520.31940.0969439740.8502883160.62550.5850437020.2492648850.37560.4729160270.3115605120.37570.7584448450.1565175230.39880.2277621560.4938219180.42590.6188914480.2230176360.352100.4425323110.4138034180.412110.9226981480.0326832980.279120.2881715150.5589600740.459130.6923516750.136288160.281140.4388306920.2976351290.375150.173718820.6570969790.45160.9148083530.0648372960.315170.374399270.464047540.466180.2211588410.6136309150.422190.5465391280.2150518280.343200.4426951860.4040242090.421100.25220.8419759760.0984019880.335230.3681060250.4741340930.478240.5785128680.2191788050.348250.2468618640.6021187950.436Next, ANFIS modeling is built based on Tables7 and 9 in MATLAB R2019b. In the developed model, there are nodes of 193, linear parameters of 405, nonlinear parameters of 36, total parameters of 441, training data of 17, testing data of 8, and fuzzy if-then rules of 81. The built ANFIS model has relatively good performance indexes with R2 of 0.963 and RMSE of 0.035.In this study, two objective functions are combined into a single objective function by using the FIS. The output of the FIS is the single objective function, which can be optimized by many methods. Then, the displacement and frequency can be predicted based on the optimal design variables. Specifically, Taguchi can optimize the single fitness function but may trap a local optimum value; meanwhile MFO algorithm can search a global optimum value. Hence, the optimal values from the Taguchi are compared with those found by the proposed method’s framework. The purpose of this comparison is to clearly demonstrate the usefulness of the devoted method in searching the global solution.In Table10, the optimal parameters by the TF include T2 = 0.45 mm, L2 = 27.5 mm, T3 = 0.54 mm, and L3 = 20 mm. The results of TF show the frequency, displacement, parasitic error, and stress are approximately 75.871 Hz and 1.776 mm, 0.019 mm, and 212.114 MPa, respectively. However, those optimal values can trap local optimum solutions. To find a global optimum solution, moth-flame optimization algorithm is adopted. The results of the hybrid computational method determine T2 = 0.45 mm, L2 = 27.5 mm, T3 = 0.78 mm, and L3 = 18 mm (Table 10). The frequency, displacement, parasitic error, and stress are about 85.174 Hz and 2.447 mm, 0.016 mm, and 145.982 MPa, respectively. Furthermore, the MPCI in the hybrid computational method is better than that in TF. It means that the hybrid computational method outperformed the TF.Table 10
Case study 1: the optimal results.
MethodOptimal parametersOptimal resultsMPCIf1 (x) (HZ)f2 (x) (mm)f3 (x) (mm)f4 (x) (MPa)n-safety factorTFT2 = 0.450.62875.87111.776360.0198212.1142.731L2 = 27.5T3 = 0.54L3 = 20The current methodT2 = 0.450.84885.17492.44720.0160145.9823.468L2 = 27.5T3 = 07.8L3 = 18


## 4.2.3. Investigation on Case Study 2

Initialized from Table5 and Figure 7, the space of design parameters is also limited to generate a new population for the modeling and optimization procedure. The optimization formulation is stated as follows.Searchx = [T2, L2, T3,]T.(8)Maximizefxcase2,(9)s.t.f1x≥1.7Hz,f2x≥1.7mm,f3x≤0.02mm,f4x≤503MPan,0.45mm≤T2≤0.65mm,22.5mm≤L2≤27.5mm,0.54mm≤T3≤0.78mm.Table11 gives the results of 13 numerical experiments, including the frequency, the displacement, the parasitic motion, and stress. The desirabilities are calculated (Table 12).Table 11
Case study 2: simulated results.
TrialRefined variables (unit: mm)FrequencyDisplacementParasiticStressT2L2T3f1 (x) (Hz)f2 (x) (mm)f3 (x) (mm)f4 (x) (MPa)10.55250.66104.04820.93360.0124162.735520.4522.50.6699.318411.04590.0155153.047230.6522.50.66127.18880.58060.008192.9515840.4527.50.6674.182281.84190.0211227.180350.6527.50.66107.04810.87420.0120117.824960.45250.5488.263161.26650.0129152.133770.65250.54101.89410.88190.0102109.092980.45250.78102.23311.00510.0130150.891690.65250.78128.56770.61220.0100105.7656100.5522.50.5498.004640.96200.0110130.6791110.5527.50.5485.376071.35840.0155158.0305120.5522.50.78123.7160.66660.0106134.808130.5527.50.7896.115231.11310.0157159.7465Table 12
Case study 2: results of the output of FIS system.
TrialD1 of f1 (x)D2 of f2 (x)Output10.5491537090.2798440080.37520.4983239310.3284312420.37530.9153048810.0894116220.34240.0593410720.9105883780.62550.568175480.2733139290.37560.2208068270.609727790.42170.5669217960.1748976850.29980.4584019210.4006139180.3959100.25100.4399921780.2769963760.375110.1845846530.6401519010.409120.9320285880.0446913770.294130.4013238530.4475952950.448Subsequently, the fuzzy if-then rules are built based on Table8, and the output of FIS system is given in Table 12.In this ANFIS model, there are nodes of 34, linear parameters of 32, nonlinear parameters of 18, total parameters of 50, training data of 9, testing data of 4, and fuzzy if-then rules of 8. The developed ANFIS modeling achieves relatively good performance indexes withR2 of 0.954 and RMSE of 0.012.Using TF, the optimal parameters areT2 = 0.45 mm, L2 = 27.5 mm, and T3 = 0.66 mm (Table 13). The frequency, displacement, parasitic error, and stress are about 79.460 Hz, 1.637 mm, 0.018 mm, and 198.015 MPa, respectively. The safety factor is determined to be about 2.54. The optimal solutions are local solutions.Table 13
Case study 2: the optimal results.
MethodOptimal parameters (mm)Optimal resultsMPCIFrequencyf1 (x) (HZ)Displacementf2 (x) (mm)Parasiticf3 (x) (mm)Stressf4 (x) (MPa)n-safety factorTFT2 = 0.450.556879.46011.63710.0184198.0152.540L2 = 27.5T3 = 0.66Computational methodT2 = 0.450.640276.74351.70020.0197236.0272.131L2 = 27.5T3 = 0.69In Table13, using the hybrid computational method, the optimal parameters are T2 = 0.45 mm, L2 = 27.5 mm, and T3 = 0.69 mm. The optimal frequency, displacement, parasitic error, equivalent stress, and safety factor are 76.743 Hz, 1.700 mm, 0.019 mm, 236.027 MPa, and 2.131, respectively. It is noted that the MPCI in the hybrid computational method is also better than that in the TF.


## 4.3. Discussion

In the previous sections, the sensitivity of factors on the output responses is analyzed to redetermine the main geometrical parameters. Those parameters contribute directly the frequency, displacement, parasitic motion, and stress of 1-DOF mechanism. Two numerical examples are studied. In order to calculate an error among the predicted value (Rp) and simulation (Rs), the error (E) is calculated as(10)E%=Rp−RsRs−1∗100.A comparison with case 1 is performed. Using the computational method, the error is around 7% for case study 1 and 3.8% for case 2. On the contrary, using TF, the error is 23% for case 1 and 14.7% for case 2 (Table14). Additionally, the computational methodology can reach a global solution. It can infer that the proposed computational scheme is greater than TF. Besides, case 1 is adopted as an optimal candidate thanks to its highest MPCI value of 0.848.Table 14
Comparison results of two cases.
Case studyParameters (mm)Optimal responsesf1 (x) (Hz)f2 (x) (mm)f3 (x) (mm)f4 (x) (MPa)nMPCICase 1The current methodT2 = 0.450.848L2 = 27.5Rp = 79.5179Rp = 1.8977Rp = 0.0224Rp = 210.382Rp = 2.390T3 = 0.78Rs = 76.062Rs = 1.7611Rs = 0.0216Rs = 223.38Rs = 2.155L3 = 18E = 4.54%E = 7.75%E = 3.70%E = 5.81%E = 1.09%Case 2The current methodT2 = 0.45Rp = 76.7435Rp = 1.8266Rp = 0.0211Rp = 236.027Rp = 2.1310.640L2 = 27.5Rs = 74.205Rs = 1.8350Rs = 0.0214Rs = 227.210Rs = 2.213T3 = 0.69E = 3.42%E = 0.45%E = 1.40%E = 3.88%E = 3.70%Figures10(a) and 10(b) illustrate the deformation and stress distributions of the mechanism, respectively.Figure 10
Simulations for case 1: (a) deformation and (b) stress distribution.
(a)(b)


## 4.4. Comparisons among Different Methods

The previous section shows that the computational method outperforms the TF in searching a global optimal solution of the 1-DOF mechanism. The comparison of the behaviors of the suggested method with other algorithms includes ANFIS-teaching-learning-based algorithm (ANFIS-coupled TLBO) [44] and ANFIS-Jaya [45]. Nonparameter statistical investigations are performed and involved in resolving case 1. In Table 15, the MPCI is almost the same for three methods. However, the output responses of the mechanism from the suggested method are better than those from the other algorithms.Table 15
Comparison among different methods.
Approachesf1 (x) (Hz)f2 (x) (mm)MPCIANFIS-Jaya (T2 = 0.45 mm, L2 = 27.49 mm, T3 = 0.77 mm, L3 = 18 mm)76.8801.7190.848ANFIS-TLBO (T2 = 0.45 mm, L2 = 27.45 mm, T3 = 0.78 mm, L3 = 18 mm)76.3761.7460.848The current method (T2 = 0.45 mm, L2 = 27.5 mm, T3 = 0.78 mm, L3 = 18 mm)79.5171.8970.848Two nonparameter statistical techniques [57] are employed to determine the behaviors of three methods. Each method has 60 runs. As given in Table 16, it is inferred that the suggested computational method is superior to other methods.Table 16
Results of Wilcoxon tests.
FrequencyThe current method and ANFIS-coupled JayaWilcoxon indexp valueDifference0≤0.001−2.6379The current method and ANFIS-coupled TLBOWilcoxon indexp valueDifference0≤0.001−3.1419DisplacementThe current method and ANFIS-coupled JayaWilcoxon indexp valueDifference0≤0.001−0.1786The current method versus ANFIS-coupled TLBOWilcoxon indexp valueDifference0≤0.001−0.1513In Table17, the results of Friedman tests also prove that the computational method outperformed the others.Table 17
Results of Friedman tests.
FrequencyMethodAverageRankANFIS-coupled Jaya76.880060ANFIS-coupled TLBO76.3760180The current method79.5179120Overall77.5913p value≤0.001DisplacementMethodAverageRankANFIS-coupled Jaya1.7191060ANFIS-coupled TLBO1.74640180The current method1.89770120Overall1.78773p value≤0.001In order to compare the convergence speed among different algorithms in the literature, the proposed method (a hybridization of desirability, fuzzy, and ANFIS-based MFO) is compared with the ant lion optimizer (ALO) [48], particle swarm optimization-based gravitational search algorithm (PSOGSA) [49], and sine-cosine algorithm (SCA) [50]. A maximum iteration of 10000 and initial population of 20 are used for all algorithms. The results found that the current method in this study has faster convergence than others because the devoted method only needs a computation time of 559.78 s, as given in Table 18.Table 18
Statistical results for comparison among metaheuristic methods.
MethodResultsTime (s)Frequency (Hz)Displacement (mm)ANFIS-coupled PSOGSA79.4211.892616.038ANFIS-coupled SCA79.4261.895599.544ANFIS-coupled ALO79.4251.893623.20ANFIS-coupled LAPO79.5171.8972707.75The current method79.5171.897559.78Additionally, the superiority of the current method in this paper is also compared with the Taguchi, the desirability, and Taguchi-fuzzy methods. It is remarked that the Taguchi method is able to search the optimal solution for each response (single optimization problem), while the others are used to solve the MOO problems for case study 1. Datasets in Table7 are used for the Taguchi, Taguchi-fuzzy, and desirability methods. The results are summarized in Table 19.Table 19
Comparison among the current method, Taguchi method, and Taguchi-fuzzy method.
MethodOptimal resultsTaguchi (single optimization)Parameters (mm)T2 = 0.65T2 = 0.45T2 = 0.65T2 = 0.65L2 = 22.5L2 = 27.5L2 = 22.5L2 = 22.5T3 = 0.78T3 = 0.54T3 = 0.78T3 = 0.66L3 = 18L3 = 22L3 = 18L3 = 20Responsef1 (x) (Hz)f2 (x) (mm)f3 (x) (mm)f4 (x) (MPa)Value139.4741.7600.00890.405Taguchi-fuzzy (MOO)Parameters (mm)T2 = 0.45, L2 = 27.5, T3 = 0.54, L3 = 20Responsef1 (x) (Hz)f2 (x) (mm)f3 (x) (mm)f4 (x) (MPa)Value75.8711.7760.019212.114Desirability (MOO)Parameters (mm)T2 = 0.65, L2 = 22.5, T3 = 0.54, L3 = 21.27Responsef1 (x) (Hz)f2 (x) (mm)f3 (x) (mm)f4 (x) (MPa)Value106.8410.9300.011120.090The current method (MOO)Parameters (mm)T2 = 0.45, L2 = 27.5, T3 = 0.78, L3 = 18Responsef1 (x) (Hz)f2 (x) (mm)f3 (x) (mm)f4 (x) (MPa)Value79.5171.8970.022210.382From the results of Table19, it can be revealed that, for optimization of a single response, the Taguchi method is a more suitable tool. By using the Taguchi, the optimal parameters are found with respect to each response but it only finds the optimum value for a single cost function, as shown in Table 19. By using the Taguchi-fuzzy method, the results searched the optimal sets of parameters for MOO but this approach is based on the Taguchi reasoning. This means that the optimal solutions can be also local values. Besides, using the desirability, the results found the optimal parameters for the MOO but this technique is based on the prediction precision of the mathematical models. Lastly, the current method (a hybridization of desirability, fuzzy, and ANFIS-based MFO methods) is a reliable technique that is superior to others because it can search the global solution for the 1-DOF mechanism.Strengths of the proposed method’s framework can be listed as follows:(i)
The behaviors of the 1-DOF mechanism are easily analyzed by linear/nonlinear FEM(ii)
Influence of the units of performances on the finally optimal solution can be ignored via the desirability method(iii)
The multiple design targets are easily converted into a single cost function through the FIS(iv)
The unknown relation among the design variables and the single cost function can be precisely approximated by ANFIS(v)
The global optimum solution of the 1-DOF mechanism can be achieved via the ANFIS-based MFOHowever, this study still has drawbacks, including the computational principle and adaptive process. The computational principle requires a variety of different methods from statistics, numerical method, intelligent method, and metaheuristics. Besides, the adaptive process needs an adaptive update of new design variables and performances have not been studied yet. Finally, the computational methods have not been automatically connected.At last, limitations of the present design framework concentrate on time and computational complexity as well as efficiency. This method needs a long period of computing time for each method. The computational complexity is mainly dependent on computer ability and experiences in the field. For a complex mechanism, the FEM and working time of computer are restricted.


## 5. Conclusions

This paper proposed the computational method and its application in design optimization of compliant mechanisms. The 1-DOF compliant mechanism is used as the study object. The suggested method is a hybridization of statistics, FEM, artificial intelligence, and metaheuristics. The usefulness of the suggested method is tested through the numerical examples and statistical comparisons. An initial 3D model in FEM is designed, and the numerical datasets are retrieved by FEA. ANOVA is used to redetermine two fine spaces of parameters, so-called populations in MFO. Based on the refined datasets, the desirabilities of the frequency and displacement are brought into the FIS system where two objective functions become a single objective function through establishment of the fuzzy if-then rules. ANFIS is then established as predictor involving the refined parameters and the output of FIS. In order to reach a global optimization, MFO algorithm is utilized to deal with the output of FIS. The results found that the frequency is 79.517 Hz and displacement is 1.897 mm for the 1-DOF mechanism. The devoted method is better than the Taguchi, Taguchi-integrated fuzzy, and desirability methods because it can search a global solution.In finding the global optimum solution, the suggested method is a better technique in comparison with the Taguchi, desirability, and Taguchi-integrated fuzzy methods. Besides, the devoted method outperforms the other metaheuristic algorithms such as TLBO and Jaya in terms of better performances. Also, the devoted method is superior to PSOGSA, SCA, ALO, and LAPO in terms of faster convergence.In future work, experiments are carried out to verify the optimized results. The current method will be extended to optimization problems with multiple constraints. Besides, the proposed method will be also considered to apply for multi-DOF compliant mechanisms.



---

*Source: 5599624-2021-03-02.xml*