# Multiresponse Parameter Optimization for the Composite Tape Winding Process Based on GRA and RSM

Authors: Qi Hong; Yaoyao Shi
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Composite tape winding is an important forming process of composite materials, which can ensure good performance of composite products. Selection and control of key process parameters in the winding process have a great influence on the properties of products, such as void content and residual stress of products. Through experimental analysis, the residual stress and void content of the composite products are not the minimum when the prepreg winding process is carried out by using the empirical process parameters. To solve this multiobjective optimization problem, experiments were conducted using the Box–Behnken design. The multiobjective optimization problem is converted to a single-objective problem using grey relational analysis (GRA). Principal component analysis (PCA) is used to quantify the relative contributions of residual stress and void content. Regression analysis of grey relational grade (GRG) based on the experimental data was used to develop a second-order GRG prediction model. The winding process parameters were optimized with response surface methodology (RSM), and the winding experiments were carried out with these parameters. The experimental results show that the best combination of process parameters yields the best GRG results with better void content and residual stress.


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## Body

## 1. Introduction

Due to good mechanical, chemical, and physical properties, composite materials are increasingly used. Advanced resin matrix composites have been widely used in the manufacture of solid rocket motor nozzles, parts of ablation and thermal protection materials, launch tubes, and special equipment for the spacecraft. They have many advantages including low density, high strength and stiffness ratios, good corrosion resistance, and ease of integral molding [1–3]. The properties of composite prepreg tape winding products depend on the properties of the materials themselves, molding process parameters, and control accuracy. Therefore, controlling the winding process and the process parameters is the key to ensuring the products have the desired properties. The main parameters in the composite prepreg tape winding process include winding temperature, tension, force, and winding speed [4–8]. An adaptive genetic algorithm with minimum product weight is proposed in [9, 10], and RSM (RSM is a statistical method to solve multivariable problems, multiple quadratic regression equations which are used to fit the functional relationship between factors and response values, and through the analysis of regression equations to find the optimal process parameters) is used to predict the strength reliability of the composite material in high-pressure hydrogen storage vessels. One study analyzed the coupling mechanism in composite material winding process parameters and presented a quadratic regression model for the interlaminar shear strength based on RSM, yielding the optimal combination of process parameters [11]. In [12], a hybrid neural network is used to determine the optimal curing time, which can ensure complete curing of composite products.One can see from the literature that prior research mainly focused on single-objective optimization, but the filament winding products have multiple objectives to be evaluated. GRA is a method of multivariate statistical analysis. In general, we want to understand the strength of a project affected by other factors in a grey system. In [13–16], GRA is introduced to solve complex relations between multiple factors and variables. GRA has been successfully applied in many engineering fields, such as welding [17, 18], high-speed machining [19–22], mechanical design of positioning platforms [23, 24], and wire electrical discharge machining [25–30].The primary work in this paper includes calculating grey correlation coefficients (GCCs) between residual stress and porosity based on experimental data. GRG can be subsequently obtained using a weighting method. A second-order prediction model for GRG is established based on experimental data with the use of RSM. The influence of process parameters on GRG is analyzed, and the optimum combination of process parameters is obtained and verified through experiments. The results in this paper show that a particular multiobjective problem can be transformed into a single-objective problem using GRA. The optimum combination of process parameters in the composite prepreg tape winding process can be subsequently determined, and the residual stress and void content of composite products can be controlled.


## 2. Experiment Procedure and Results

### 2.1. Composite Prepreg Tape Winding Process

In the process of composite tape winding, tension is a key process parameter. Tension is applied with a magnetic powder brake during the winding process. Tension can straighten the prepreg winding so that the fiber can bear the load evenly. The applied tension can remove air bubbles, make the winding products more compact, and facilitate resin penetration. In the winding process, the core mold rotates at a constant speed, and the resin is heated to the melting state by the hot pressure roller and the hot blower. As shown in Figure1, heat comes from the internal heating wire in the hot pressure roller, which helps reduce the resin speed and increase the degree of contact between the layers. In the melting zone, the hot pressure roller exerts positive force on the tape layer, causing the prepreg tape and the tape layer to come into close contact. On the contrary, it helps reduce void content by reducing bubbles between interlayer contacts.Figure 1
Process schematic of composite tape winding.The winding of the prepreg tape is essentially a process of continuous fusion between the prepreg tape and the base layer. When a certain winding pressure and temperature are applied to the winding layer and prepreg, the surface geometry will be deformed, as shown in Figure2. The self-bonding process begins when the winding layer is in close contact with the prepreg surface. After a period of time, the polymer chain penetrates completely and entangles with the adjacent interface so that the matrix on the interface forms the bulk polymer again, and the two surfaces are fully integrated.Figure 2
Schematic of microcosmic deformation on the composite prepreg tape surface.


### 2.2. Experimental Procedures

A KUKA robot (XGD-1200) (KUKA, Augsburg, Germany) is the primary piece of equipment used in our experiments, as shown in Figure3. In order to ensure the control accuracy of the winding process, a high-precision deviation correction control system and layered superposition winding are used in the experiment. Circumferential winding experiment is carried out with the orthogonal prepreg tape. T300/epoxy orthogonal prepreg tape is purchased from Daobo Composites Co., Ltd., Xi’an, China. The winding mandrel is a 45 steel cylinder which is 150 mm in external diameter and 1200 mm in length. The parameters of the prepreg tape are fiber volume fraction, tape width, and tape thickness, which are 56 ± 2%, 80 mm, and 0.25 mm, respectively. In this experiment, the ambient temperature is 20 ± 2°C, and the relative humidity is 25 ± 2%. The curing process has a great influence on the properties of the products. During the curing process, the heating rate is kept at 2.5°C/min. When the curing temperature reaches 150°C, it is needed to keep heating for 150 minutes, and the curing pressure is 0.15 MPa.Figure 3
Composite tape winding robot (XGD-1200).
(a)(b)


### 2.3. Sample Preparation and Measurement Method

#### 2.3.1. Residual Stress

In the process of composite prepreg winding, due to the deformation of the product, the residual stress is inevitable. Residual stress is a key index to evaluate the properties of composite products, which is selected as one of the optimization objectives. When examining residual stress in composite annular parts, a slot is cut along the radial direction, and the slot position will change due to the residual stress moment. Figure4 shows the change in slot displacement after closure of the compound ring, which can be measured with an electron microscope. The circumferential residual stress in the composite ring can be calculated with the following equations [31, 32]:(1)σθ=−4Mrk−Ra2Rb2r2lnRbRa+Rb2lnrRb+a2lnRar+Rb2−Ra2,(2)Mr=−ΔE8πRb2−Ra22−4Ra2Rb2lnRb/Ra22Rb2−Ra2,k=Rb2−Ra22−4Ra2Rb2lnRb/Ra2,Δ=Δ1+Δ2,where r is the current position, Δ is the total displacement of the slot, Mr is the bending moment per unit width, E is the circumferential elastic modulus, and Ra and Rb are the inner and outer radii, respectively.Figure 4
Residual stress measurement method.The magnitude of the residual stress is the average value of the composite outer and inner rings, which can be defined as follows:(3)σΔ=σθr=Ra+σθr=Rb2.


#### 2.3.2. Void Content

Voids in composite tape winding products are defects that principally exist between tape layers due to residual air bubbles, resin flow, and sufficient compaction during resin flow. Density measurements, microscopy [33–39], ultrasonic attenuation [40], and X-ray computed tomography [41] can be used to measure void content. The electronic micrograph method has the highest porosity detection accuracy, which is specified by GB/T3365-2008 [42]. The specimen is sampled and observed at the marked position, as shown in Figure 5. The samples are cut on the composite ring product, with a length, width, and height of 20 mm, 10 mm, and 10 mm, respectively. The samples are ground and polished under flowing water, then the polished specimen is observed under a microscope, and the process of measuring void content is shown in Figure 6. The calculation formula of void content is shown in the following equation:(4)X=SvA×100%,where X is the void content, A is the cross-sectional area, and Sv is the total void area of the sample.Figure 5
Sketch of cutting the testing sample.Figure 6
Void content measurement process of composite products.


### 2.4. Experiment Results

Temperature, winding speed, roller force, and winding tension are taken as the independent experimental variables, and residual stress and porosity are the response variables and design variable values according to actual working conditions. The four-factor and three-level BBD experiment design method was used to reduce costs. BBD is a response surface design type, which does not include embedding factor or partial factor design. BBD usually has fewer design points, and each factor always has 3 levels. According to the actual production experience, the temperature setting range is 50–100°C. When the prepreg tape temperature is lower than 50°C, the resin matrix is difficult to reach the melting state, and when the temperature is too high, it will easily lead to early curing of the resin. The pressure setting range is 1000–2000 N; when the pressure is less than 1000 N, the contact between the laminate and prepreg tape is not enough, and when the pressure is too large, the winding products will be deformed. The setting range of tension is 100–500 N; too small tension will increase the number of air bubbles between layers, and too large tension will lead to the fracture of the prepreg tape. The speed setting range is 5–15 rpm, and too fast winding speed is not conducive to machining. The designed level of process parameters is shown in Table1, and the experimental results are shown in Table 2.Table 1
Level of process parameters.
Experimental parametersSymbolUnitLevel of experimental parametersLevel 1Level 2Level 3TemperatureA°C5075100TensionBN100300500ForceCN100015002000VelocityDrpm51015Table 2
Experiment and measurement results.
No.Process parametersObjective valueABCDResidual stress (MPa)Void content (%)110010015001012.120.9727550015001520.051.5637530015001013.260.3347530015001013.530.4457510010001011.971.5867510015001515.310.9877530020001518.941.06810030015001516.980.5897550020001018.851.0110755001500519.710.331110030020001015.190.5512753001000516.821.1913751001500513.721.011410030010001014.741.26157510020001012.960.89167530015001013.280.45175030010001014.12.12187550010001017.141.96197530015001012.830.3520753002000517.540.65217530010001517.741.94225010015001010.741.19237530015001013.290.4724503001500517.732.19251003001500517.030.2726503001500515.980.81275030020001015.131.592810050015001017.120.33295050015001017.711.65


## 2.1. Composite Prepreg Tape Winding Process

In the process of composite tape winding, tension is a key process parameter. Tension is applied with a magnetic powder brake during the winding process. Tension can straighten the prepreg winding so that the fiber can bear the load evenly. The applied tension can remove air bubbles, make the winding products more compact, and facilitate resin penetration. In the winding process, the core mold rotates at a constant speed, and the resin is heated to the melting state by the hot pressure roller and the hot blower. As shown in Figure1, heat comes from the internal heating wire in the hot pressure roller, which helps reduce the resin speed and increase the degree of contact between the layers. In the melting zone, the hot pressure roller exerts positive force on the tape layer, causing the prepreg tape and the tape layer to come into close contact. On the contrary, it helps reduce void content by reducing bubbles between interlayer contacts.Figure 1
Process schematic of composite tape winding.The winding of the prepreg tape is essentially a process of continuous fusion between the prepreg tape and the base layer. When a certain winding pressure and temperature are applied to the winding layer and prepreg, the surface geometry will be deformed, as shown in Figure2. The self-bonding process begins when the winding layer is in close contact with the prepreg surface. After a period of time, the polymer chain penetrates completely and entangles with the adjacent interface so that the matrix on the interface forms the bulk polymer again, and the two surfaces are fully integrated.Figure 2
Schematic of microcosmic deformation on the composite prepreg tape surface.


## 2.2. Experimental Procedures

A KUKA robot (XGD-1200) (KUKA, Augsburg, Germany) is the primary piece of equipment used in our experiments, as shown in Figure3. In order to ensure the control accuracy of the winding process, a high-precision deviation correction control system and layered superposition winding are used in the experiment. Circumferential winding experiment is carried out with the orthogonal prepreg tape. T300/epoxy orthogonal prepreg tape is purchased from Daobo Composites Co., Ltd., Xi’an, China. The winding mandrel is a 45 steel cylinder which is 150 mm in external diameter and 1200 mm in length. The parameters of the prepreg tape are fiber volume fraction, tape width, and tape thickness, which are 56 ± 2%, 80 mm, and 0.25 mm, respectively. In this experiment, the ambient temperature is 20 ± 2°C, and the relative humidity is 25 ± 2%. The curing process has a great influence on the properties of the products. During the curing process, the heating rate is kept at 2.5°C/min. When the curing temperature reaches 150°C, it is needed to keep heating for 150 minutes, and the curing pressure is 0.15 MPa.Figure 3
Composite tape winding robot (XGD-1200).
(a)(b)


## 2.3. Sample Preparation and Measurement Method

### 2.3.1. Residual Stress

In the process of composite prepreg winding, due to the deformation of the product, the residual stress is inevitable. Residual stress is a key index to evaluate the properties of composite products, which is selected as one of the optimization objectives. When examining residual stress in composite annular parts, a slot is cut along the radial direction, and the slot position will change due to the residual stress moment. Figure4 shows the change in slot displacement after closure of the compound ring, which can be measured with an electron microscope. The circumferential residual stress in the composite ring can be calculated with the following equations [31, 32]:(1)σθ=−4Mrk−Ra2Rb2r2lnRbRa+Rb2lnrRb+a2lnRar+Rb2−Ra2,(2)Mr=−ΔE8πRb2−Ra22−4Ra2Rb2lnRb/Ra22Rb2−Ra2,k=Rb2−Ra22−4Ra2Rb2lnRb/Ra2,Δ=Δ1+Δ2,where r is the current position, Δ is the total displacement of the slot, Mr is the bending moment per unit width, E is the circumferential elastic modulus, and Ra and Rb are the inner and outer radii, respectively.Figure 4
Residual stress measurement method.The magnitude of the residual stress is the average value of the composite outer and inner rings, which can be defined as follows:(3)σΔ=σθr=Ra+σθr=Rb2.


### 2.3.2. Void Content

Voids in composite tape winding products are defects that principally exist between tape layers due to residual air bubbles, resin flow, and sufficient compaction during resin flow. Density measurements, microscopy [33–39], ultrasonic attenuation [40], and X-ray computed tomography [41] can be used to measure void content. The electronic micrograph method has the highest porosity detection accuracy, which is specified by GB/T3365-2008 [42]. The specimen is sampled and observed at the marked position, as shown in Figure 5. The samples are cut on the composite ring product, with a length, width, and height of 20 mm, 10 mm, and 10 mm, respectively. The samples are ground and polished under flowing water, then the polished specimen is observed under a microscope, and the process of measuring void content is shown in Figure 6. The calculation formula of void content is shown in the following equation:(4)X=SvA×100%,where X is the void content, A is the cross-sectional area, and Sv is the total void area of the sample.Figure 5
Sketch of cutting the testing sample.Figure 6
Void content measurement process of composite products.


## 2.3.1. Residual Stress

In the process of composite prepreg winding, due to the deformation of the product, the residual stress is inevitable. Residual stress is a key index to evaluate the properties of composite products, which is selected as one of the optimization objectives. When examining residual stress in composite annular parts, a slot is cut along the radial direction, and the slot position will change due to the residual stress moment. Figure4 shows the change in slot displacement after closure of the compound ring, which can be measured with an electron microscope. The circumferential residual stress in the composite ring can be calculated with the following equations [31, 32]:(1)σθ=−4Mrk−Ra2Rb2r2lnRbRa+Rb2lnrRb+a2lnRar+Rb2−Ra2,(2)Mr=−ΔE8πRb2−Ra22−4Ra2Rb2lnRb/Ra22Rb2−Ra2,k=Rb2−Ra22−4Ra2Rb2lnRb/Ra2,Δ=Δ1+Δ2,where r is the current position, Δ is the total displacement of the slot, Mr is the bending moment per unit width, E is the circumferential elastic modulus, and Ra and Rb are the inner and outer radii, respectively.Figure 4
Residual stress measurement method.The magnitude of the residual stress is the average value of the composite outer and inner rings, which can be defined as follows:(3)σΔ=σθr=Ra+σθr=Rb2.


## 2.3.2. Void Content

Voids in composite tape winding products are defects that principally exist between tape layers due to residual air bubbles, resin flow, and sufficient compaction during resin flow. Density measurements, microscopy [33–39], ultrasonic attenuation [40], and X-ray computed tomography [41] can be used to measure void content. The electronic micrograph method has the highest porosity detection accuracy, which is specified by GB/T3365-2008 [42]. The specimen is sampled and observed at the marked position, as shown in Figure 5. The samples are cut on the composite ring product, with a length, width, and height of 20 mm, 10 mm, and 10 mm, respectively. The samples are ground and polished under flowing water, then the polished specimen is observed under a microscope, and the process of measuring void content is shown in Figure 6. The calculation formula of void content is shown in the following equation:(4)X=SvA×100%,where X is the void content, A is the cross-sectional area, and Sv is the total void area of the sample.Figure 5
Sketch of cutting the testing sample.Figure 6
Void content measurement process of composite products.


## 2.4. Experiment Results

Temperature, winding speed, roller force, and winding tension are taken as the independent experimental variables, and residual stress and porosity are the response variables and design variable values according to actual working conditions. The four-factor and three-level BBD experiment design method was used to reduce costs. BBD is a response surface design type, which does not include embedding factor or partial factor design. BBD usually has fewer design points, and each factor always has 3 levels. According to the actual production experience, the temperature setting range is 50–100°C. When the prepreg tape temperature is lower than 50°C, the resin matrix is difficult to reach the melting state, and when the temperature is too high, it will easily lead to early curing of the resin. The pressure setting range is 1000–2000 N; when the pressure is less than 1000 N, the contact between the laminate and prepreg tape is not enough, and when the pressure is too large, the winding products will be deformed. The setting range of tension is 100–500 N; too small tension will increase the number of air bubbles between layers, and too large tension will lead to the fracture of the prepreg tape. The speed setting range is 5–15 rpm, and too fast winding speed is not conducive to machining. The designed level of process parameters is shown in Table1, and the experimental results are shown in Table 2.Table 1
Level of process parameters.
Experimental parametersSymbolUnitLevel of experimental parametersLevel 1Level 2Level 3TemperatureA°C5075100TensionBN100300500ForceCN100015002000VelocityDrpm51015Table 2
Experiment and measurement results.
No.Process parametersObjective valueABCDResidual stress (MPa)Void content (%)110010015001012.120.9727550015001520.051.5637530015001013.260.3347530015001013.530.4457510010001011.971.5867510015001515.310.9877530020001518.941.06810030015001516.980.5897550020001018.851.0110755001500519.710.331110030020001015.190.5512753001000516.821.1913751001500513.721.011410030010001014.741.26157510020001012.960.89167530015001013.280.45175030010001014.12.12187550010001017.141.96197530015001012.830.3520753002000517.540.65217530010001517.741.94225010015001010.741.19237530015001013.290.4724503001500517.732.19251003001500517.030.2726503001500515.980.81275030020001015.131.592810050015001017.120.33295050015001017.711.65


## 3. Grey Relational Grade Calculation from Experimental Data

### 3.1. GRG Calculation Process

Step 1: experimental void content and residual stress data were collected and used to carry out a standardized treatment. Composite tape winding products with lower void content and residual stress are desirable. Therefore, the experimental data can be expressed as follows:(5)xi∗k=maxxiok−xiokmaxxiok−minxiok,where xiok is the original sequence, xi∗k is the comparison sequence, k = 1, 2, …, n, i = 1, 2, …, m, and n and m are the total number of response variables and experimental runs.Step 2: the grey correlation coefficient can be calculated by the following equation:(6)γx0∗k,xi∗k=Δmin+ζΔmaxΔ0ik+ζΔmax,Δ0ik=x0∗k−xi∗k,Δmax=maximaxjΔ0ik,Δmin=miniminjΔ0ik,where Δ0ik is the deviation sequence, x0∗k is the reference sequence, and ζ∈0,1 is the distinguishing coefficient.Step 3: calculation of the weight value for response variables: in this paper, the contributions of void content and residual stress are quantified by PCA. The calculation process is expressed as follows:(a)
Establishing the original sequence of various quality characteristics:(7)x=x11x12⋯x1nx21x22⋯x2n⋮⋮⋮⋮xm1xm2⋯xmn,where m is the number of experiments, n is the number of response variables, and x is the grey correlation coefficient of each response variable.(b)
Calculation of the correlation coefficient array:(8)Rjl=Covxij,xilσxij×σxil,where Covxij,xil is the covariance of xij and xil and σxij and σxil are the standard deviations of xij and xil.(c)
Computation of eigenvalues and eigenvectors:λk can be obtained by the following equation:(9)λkIm−R=0,where Im is the identity matrix and the eigenvalues λk are arranged in the ascending order, i.e., λ1≥λ2≥⋯≥0, k = 1, 2, …, n.(d)
Calculation of weight of the principal component:(10)ak=λk∑λii=1n,Nk=∑λki=1k∑λii=1n.Generally, λ1 and λ2 are called the first and second principal components if Nk>85%, respectively.Step 4: calculation of the grey relational grade: GRG is the weighted sum of the grey relational coefficient and can be calculated by the following equation:(11)γx0∗,xi∗=∑k=1nβkx0∗k,xi∗k,where ∑k=1nβk=1, and βk is the weight of the kth response variable and determined by PCA.


### 3.2. Results for GRA

Principal component analysis for weight values and results for GRA are provide in Tables3 and 4.Table 3
Principal component analysis for weight values.
Principal componentEigenvalueContribution (%)Residual stress1.101355.1Void content0.898744.9Total100Table 4
Results for GRA.
No.Comparison sequenceDeviation sequenceGRCGRGxi∗RSxi∗VCΔ0i (RS)Δ0i (VC)GRC (RS)GRC (VC)10.8517720.6559140.1482280.3440860.7713340.5923570.690973200.33870910.6612900.3333330.4305550.37698630.72932310.27067600.64878010.80647840.7003220.9408600.2996770.0591390.6252510.8942300.74602350.8678830.3279560.1321160.6720430.7909940.4266050.62738360.5091290.6505370.4908700.3494620.5046070.5886070.66232370.1192260.6075260.8807730.3924730.3621150.5602400.45107480.3297520.8655910.6702470.1344080.4272600.7881350.58929390.1288930.6344080.8711060.3655910.364660.5776390.500292100.03651910.96348000.34165110.607724110.5220190.8817200.4779800.1182790.5112570.8086950.644807120.3469380.5376340.6530610.4623650.4336280.5195530.472208130.6799140.6344080.3200850.3655910.6096920.5776390.595300140.5703540.50.4296450.50.5378390.50.520849150.7615460.6989240.2384530.3010750.6770900.6241610.653325160.7271750.9354830.2728240.0645160.6469770.8857140.754170170.6390970.0376340.3609020.9623650.5807860.3419110.473531180.3125670.2688170.6874320.7311820.4210760.4061130.414358190.7755100.9892470.2244890.0107520.3347690.9789470.819814200.2696020.8279560.7303970.1720430.4063720.7440.557967210.2481200.1344080.7518790.8655910.3993990.3661410.3844662210.53763400.46236510.5195530.784279230.7261000.9247310.2738990.0752680.6460790.8691580.746241240.24919400.75080510.3997420.3333330.369924250.3243820.7634400.6756170.2365590.4253080.6788320.539140260.4371640.7419350.5628350.2580640.4704390.6595740.555361270.5284640.3225800.4715350.6774190.5146480.4246570.474242280.31471510.68528400.42183910.681433290.2513420.2903220.7486570.7096770.4004300.4133330.406223Notes: Δ0i: deviation sequence; xi∗: comparison sequence; GRC: grey relational coefficient; RS: residual stress; and VC: void content.


## 3.1. GRG Calculation Process

Step 1: experimental void content and residual stress data were collected and used to carry out a standardized treatment. Composite tape winding products with lower void content and residual stress are desirable. Therefore, the experimental data can be expressed as follows:(5)xi∗k=maxxiok−xiokmaxxiok−minxiok,where xiok is the original sequence, xi∗k is the comparison sequence, k = 1, 2, …, n, i = 1, 2, …, m, and n and m are the total number of response variables and experimental runs.Step 2: the grey correlation coefficient can be calculated by the following equation:(6)γx0∗k,xi∗k=Δmin+ζΔmaxΔ0ik+ζΔmax,Δ0ik=x0∗k−xi∗k,Δmax=maximaxjΔ0ik,Δmin=miniminjΔ0ik,where Δ0ik is the deviation sequence, x0∗k is the reference sequence, and ζ∈0,1 is the distinguishing coefficient.Step 3: calculation of the weight value for response variables: in this paper, the contributions of void content and residual stress are quantified by PCA. The calculation process is expressed as follows:(a)
Establishing the original sequence of various quality characteristics:(7)x=x11x12⋯x1nx21x22⋯x2n⋮⋮⋮⋮xm1xm2⋯xmn,where m is the number of experiments, n is the number of response variables, and x is the grey correlation coefficient of each response variable.(b)
Calculation of the correlation coefficient array:(8)Rjl=Covxij,xilσxij×σxil,where Covxij,xil is the covariance of xij and xil and σxij and σxil are the standard deviations of xij and xil.(c)
Computation of eigenvalues and eigenvectors:λk can be obtained by the following equation:(9)λkIm−R=0,where Im is the identity matrix and the eigenvalues λk are arranged in the ascending order, i.e., λ1≥λ2≥⋯≥0, k = 1, 2, …, n.(d)
Calculation of weight of the principal component:(10)ak=λk∑λii=1n,Nk=∑λki=1k∑λii=1n.Generally, λ1 and λ2 are called the first and second principal components if Nk>85%, respectively.Step 4: calculation of the grey relational grade: GRG is the weighted sum of the grey relational coefficient and can be calculated by the following equation:(11)γx0∗,xi∗=∑k=1nβkx0∗k,xi∗k,where ∑k=1nβk=1, and βk is the weight of the kth response variable and determined by PCA.


## 3.2. Results for GRA

Principal component analysis for weight values and results for GRA are provide in Tables3 and 4.Table 3
Principal component analysis for weight values.
Principal componentEigenvalueContribution (%)Residual stress1.101355.1Void content0.898744.9Total100Table 4
Results for GRA.
No.Comparison sequenceDeviation sequenceGRCGRGxi∗RSxi∗VCΔ0i (RS)Δ0i (VC)GRC (RS)GRC (VC)10.8517720.6559140.1482280.3440860.7713340.5923570.690973200.33870910.6612900.3333330.4305550.37698630.72932310.27067600.64878010.80647840.7003220.9408600.2996770.0591390.6252510.8942300.74602350.8678830.3279560.1321160.6720430.7909940.4266050.62738360.5091290.6505370.4908700.3494620.5046070.5886070.66232370.1192260.6075260.8807730.3924730.3621150.5602400.45107480.3297520.8655910.6702470.1344080.4272600.7881350.58929390.1288930.6344080.8711060.3655910.364660.5776390.500292100.03651910.96348000.34165110.607724110.5220190.8817200.4779800.1182790.5112570.8086950.644807120.3469380.5376340.6530610.4623650.4336280.5195530.472208130.6799140.6344080.3200850.3655910.6096920.5776390.595300140.5703540.50.4296450.50.5378390.50.520849150.7615460.6989240.2384530.3010750.6770900.6241610.653325160.7271750.9354830.2728240.0645160.6469770.8857140.754170170.6390970.0376340.3609020.9623650.5807860.3419110.473531180.3125670.2688170.6874320.7311820.4210760.4061130.414358190.7755100.9892470.2244890.0107520.3347690.9789470.819814200.2696020.8279560.7303970.1720430.4063720.7440.557967210.2481200.1344080.7518790.8655910.3993990.3661410.3844662210.53763400.46236510.5195530.784279230.7261000.9247310.2738990.0752680.6460790.8691580.746241240.24919400.75080510.3997420.3333330.369924250.3243820.7634400.6756170.2365590.4253080.6788320.539140260.4371640.7419350.5628350.2580640.4704390.6595740.555361270.5284640.3225800.4715350.6774190.5146480.4246570.474242280.31471510.68528400.42183910.681433290.2513420.2903220.7486570.7096770.4004300.4133330.406223Notes: Δ0i: deviation sequence; xi∗: comparison sequence; GRC: grey relational coefficient; RS: residual stress; and VC: void content.


## 4. The Grey Relational Grade Prediction Model

According to PCA for weight values as shown in Table3 and results for GRA as shown in Table 4, a mapping relation between the process parameters and the GRG can be established. Generally, a second-order mathematical regression model in response surface methodology is used to determine the relationship between the response variable and the input factor. The second-order GRG model for the winding process parameters can be expressed as follows:(12)y^=GRG−ε=ρ0+∑i=14ρiyi+∑i=14∑j=i+14ρijyiyj+∑i=14ρiiyi2,where y^ is the estimated GRG value, yi is a winding process parameter, ε is the experimental error, and each ρ is a second-order regression coefficient. The second term models a linear effect, the third term models an interaction effect, and the fourth term models second-order effects.GRG prediction model is based on regression analysis of experimental data using Minitab software. A comparison between the predicted and calculated GRG values is shown in Figure7. Figure 7 shows that the predicted GRG value is very close to the calculated value, with an average deviation of 1.22%. Thus, there is no significant difference between the predicted value and the calculated value. The residual error in the prediction model is shown in Figure 8, which shows that residual errors are randomly distributed near zero without abnormal points. Thus, the prediction model is a good fit to the experimental data. Analysis of variance (ANOVA) results for the prediction model are shown in Table 5. One can see from the results in Table 5 that the coefficients of determination (R-Sq and R-Sq (adj)) are very close, indicating that the model is very reliable and accurate. The prediction model is shown in the following equation:(13)y^=−1.395+0.00945A−0.000461B+0.001756C+0.1097D−0.000143A2−0.000001B2−0.000001C2−0.006405D2+0.000018AB+0.000002AC+0.000471AD−0.000074BD−0.000002CD.Figure 7
Comparison chart of the experimental value and the predicted value of GRG.Figure 8
Residual plot for the GRG prediction model.Table 5
ANOVA for the response surface model.
SourceDFSSMSFF0.01Regression model140.5128810.03663452.263.698Error140.0098130.000701Total280.522695Standard deviationR-Sq = 98.12% R-Sq (adj) = 96.25%Notes: DF: degree of freedom; SS: sum of square; MS: mean square; and SD: standard deviation.


## 5. Multiresponse Parameter Optimization and Experimental Verification

### 5.1. Analysis on Parameter Influence Laws

If the GRG is larger, the response variable will be better. Correspondingly, when the average GRG value for each process parameter is the largest, the response variable is the best. Table6 shows that the optimal level of temperature is level 3 (100°C), the optimal level of tension is level 1 (100 N), the optimal level of force is level 2 (1500 N), and the optimal level of rotation speed is level 2 (10 rpm). The difference between minimum and maximum values shows that tension has the greatest impact on the multiobjective response followed by the influence of speed, force, and temperature. Figure 9 also shows that tension has the greatest impact on residual stress, while the impact of temperature is minimal. The aforementioned parameter combination yields products with minimum residual stress.Table 6
The average grey relational grade of each parameter.
Process parametersTemperatureTensionForceSpeedLevel 10.51050.66890.48210.5446Level 20.59850.58260.63120.6320Level 30.61100.49780.54690.4723Max-min0.09050.17010.14910.1597Figure 9
Main effect plot of GCC (residual stress).Figure10 shows that temperature and force have a great impact on void content, while the impact of tension is minimal. The parameter combination yielding products with minimum void content is 50°C temperature, 100 N tension, 1000 N force, and 15 rpm rotation speed. The influence of process parameters on GRG is given by the weighted sum of residual stress and void content. Therefore, the influence of process parameters on residual stress and void content can be determined by examining changes in the GRG value, and residual stress and void content can be optimized by optimizing GRG.Figure 10
Main effect plot of GCC (void content).


### 5.2. Optimal Parameters

The GRG prediction model is analyzed using the response optimizer in Minitab software. Figure11 shows that the optimal GRG is 0.8106 as calculated with the weighting method, and the optimal parameter combination is 70.20°C temperature, 100.08 N tension, 1515.15 N force, and 10.25 rpm rotation speed. Figure 12 shows that the optimal GRG value is 0.8077 when calculated with the equal weight method, and the optimal parameter combination is 78.789°C temperature, 237.37 N tension, 1555.55 N force, and 9.545 rpm rotation speed.Figure 11
Optimal process parameter combination (weighting method).Figure 12
Optimal process parameters (equal weight method).


### 5.3. Experimental Verification

Once the optimum combination of process parameters is determined, a confirmation experiment is conducted to validate the optimal solution. The results of the eighth experiment are used as the comparative group. Table7 shows the comparison of winding experimental results between the initial parameter values and the optimum process parameters. The results show that the proposed method can effectively reduce the residual stress and the void content of the composite winding product.Table 7
Comparison between the initial and optimum process parameters.
Initial process parameter valueOptimal process conditionImprovement (%)PredictionExperimental valueTemperature (°C)7570.0270.02Tension (N)300100.08100.08Force (N)15001515.151515.15Velocity (rpm)1010.2510.25Residual stress (MPa)13.2911.1216.3Void content (%)0.470.3819.1GRG0.74620.81060.81238.8The winding experiment is carried out with parameter combination calculated by the weighting method and parameter combination calculated by the equal weight method. Table8 shows that the equal weight optimization parameter combination produces lower residual stress, but the void content in the product is greater than 0.95%. In order to ensure the mechanical properties of composite products, the void content should be kept at a low value, especially for special aeronautical parts. The experimental results show that the weighted method is better than the equal weight method in the winding process.Table 8
Experimental results of different process parameters’ combination.
Experimental schemeABCDResidual stress (MPa)Void content (%)Equal weight method78.78237.315559.5459.680.96Weighting method70.02100.08151510.2511.120.38


## 5.1. Analysis on Parameter Influence Laws

If the GRG is larger, the response variable will be better. Correspondingly, when the average GRG value for each process parameter is the largest, the response variable is the best. Table6 shows that the optimal level of temperature is level 3 (100°C), the optimal level of tension is level 1 (100 N), the optimal level of force is level 2 (1500 N), and the optimal level of rotation speed is level 2 (10 rpm). The difference between minimum and maximum values shows that tension has the greatest impact on the multiobjective response followed by the influence of speed, force, and temperature. Figure 9 also shows that tension has the greatest impact on residual stress, while the impact of temperature is minimal. The aforementioned parameter combination yields products with minimum residual stress.Table 6
The average grey relational grade of each parameter.
Process parametersTemperatureTensionForceSpeedLevel 10.51050.66890.48210.5446Level 20.59850.58260.63120.6320Level 30.61100.49780.54690.4723Max-min0.09050.17010.14910.1597Figure 9
Main effect plot of GCC (residual stress).Figure10 shows that temperature and force have a great impact on void content, while the impact of tension is minimal. The parameter combination yielding products with minimum void content is 50°C temperature, 100 N tension, 1000 N force, and 15 rpm rotation speed. The influence of process parameters on GRG is given by the weighted sum of residual stress and void content. Therefore, the influence of process parameters on residual stress and void content can be determined by examining changes in the GRG value, and residual stress and void content can be optimized by optimizing GRG.Figure 10
Main effect plot of GCC (void content).


## 5.2. Optimal Parameters

The GRG prediction model is analyzed using the response optimizer in Minitab software. Figure11 shows that the optimal GRG is 0.8106 as calculated with the weighting method, and the optimal parameter combination is 70.20°C temperature, 100.08 N tension, 1515.15 N force, and 10.25 rpm rotation speed. Figure 12 shows that the optimal GRG value is 0.8077 when calculated with the equal weight method, and the optimal parameter combination is 78.789°C temperature, 237.37 N tension, 1555.55 N force, and 9.545 rpm rotation speed.Figure 11
Optimal process parameter combination (weighting method).Figure 12
Optimal process parameters (equal weight method).


## 5.3. Experimental Verification

Once the optimum combination of process parameters is determined, a confirmation experiment is conducted to validate the optimal solution. The results of the eighth experiment are used as the comparative group. Table7 shows the comparison of winding experimental results between the initial parameter values and the optimum process parameters. The results show that the proposed method can effectively reduce the residual stress and the void content of the composite winding product.Table 7
Comparison between the initial and optimum process parameters.
Initial process parameter valueOptimal process conditionImprovement (%)PredictionExperimental valueTemperature (°C)7570.0270.02Tension (N)300100.08100.08Force (N)15001515.151515.15Velocity (rpm)1010.2510.25Residual stress (MPa)13.2911.1216.3Void content (%)0.470.3819.1GRG0.74620.81060.81238.8The winding experiment is carried out with parameter combination calculated by the weighting method and parameter combination calculated by the equal weight method. Table8 shows that the equal weight optimization parameter combination produces lower residual stress, but the void content in the product is greater than 0.95%. In order to ensure the mechanical properties of composite products, the void content should be kept at a low value, especially for special aeronautical parts. The experimental results show that the weighted method is better than the equal weight method in the winding process.Table 8
Experimental results of different process parameters’ combination.
Experimental schemeABCDResidual stress (MPa)Void content (%)Equal weight method78.78237.315559.5459.680.96Weighting method70.02100.08151510.2511.120.38


## 6. Conclusion

(1)
The optimization of composite tape winding process parameters is a multiparameter input and multiobjective response. A procedure integrating the Box–Behnken design, RSM, and GRA is used to predict the optimal process conditions for reducing the residual stress and void content in composite products.(2)
The GRG prediction model was established using Box–Behnken design based on experimental data. ANOVA results show that the prediction model is reliable and significant. The optimal combination of process parameters was found to be 70.02°C temperature, 100.08 N tension, 1515.15 N force, and 10.25 rpm winding speed. The predicted results are in good agreement with the experimental results. Residual stress and void content can be improved by using optimized parameter combination to carry out the winding process.(3)
Comparing the experimental results of weighted optimal parameter combination and equal weight optimal parameter combination, the experimental results show that GRA based on equal weight has some limitations in optimizing the parameters of the composite tape winding process, and the weighted GRA method proposed in this paper has obvious advantages.



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*Source: 2515014-2020-07-15.xml*