# Fabrication of High-Quality Polymer Composite Frame by a New Method of Fiber Winding Process

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## Abstract

**:**

## 1. Introduction

## 2. Manufacturing of Polymer Composite Frame

#### 2.1. Fiber Winding Geometry

_{3}is taken into account. Vectors and matrices are written in a homogeneous form (i.e., any point $V={\left[{x}_{V},{y}_{V},{z}_{V},1\right]}^{T}$ and any vector $u={\left({x}_{u},{y}_{u},{z}_{u},0\right)}^{T}$, in more detail see [38]). The Euclidean norm $\Vert u\Vert $ of vector $u$, where $\Vert u\Vert =\sqrt{{x}_{u}^{2}+{y}_{u}^{2}+{z}_{u}^{2}}$ is used.

_{3}and radius $r$ is considered. Then arms of oriented angles ${\lambda}_{i}$ create on the circle the vertices ${A}_{i}={(r\mathrm{cos}{\lambda}_{i},\text{}r\mathrm{sin}{\lambda}_{i},\text{}0,\text{}1)}^{T}$ of a regular twelve-rectangle (Figure 5 (left)).

#### 2.2. Mathematical Model of Winding Process

_{3}of the robot ($BCS$) is taken into account. This system is often called the “robot coordinate system” for industrial robots (Figure 7 (right)). Individual parts of the mathematical winding model are described in $BCS$.

_{3}of the REE ($LCS$) is also taken into account. The position of $LCS$ toward $BCS$ defines position of REE in $BCS$.

#### 2.2.1. Fiber-Processing Head

#### 2.2.2. Non-Bearing Core Frame

#### 2.3. Robot Trajectory Optimization

**Note 1**

**Note 2**

#### 2.3.1. Schematic Representation of the Procedure for Calculating the Optimal REE Trajectory

**Note 3**(based on the flowchart shown in Figure 11)

- Specification of the fiber-processing head in $BCS$ (including coordinates of centers $S{1}_{BCS}$ and $S{2}_{BCS}$ of outer rotating guide lines $k1$ and $k2$ of the head, vectors $h1{}_{BCS}$ and $h2{}_{BCS}$, common radius ${r}_{CIRCLE}$ of circles $k1$ and $k2$).
- Loading of the location of composite frame in $LCS$ (including coordinates of points $B{(i)}_{LCS}$, vectors $b1\text{}{(i)}_{LCS}$, $b2\text{}{(i)}_{LCS}$ and values $C{(i)}_{}$ for $1\le i\le N$, radius of frame ${r}_{TUBE}$).
- Determination of more $B{(i)}_{LCS}$ points on frame axis ${o}_{LCS}$ and corresponding vectors $b1\text{}{(i)}_{LCS}$ and $b2\text{}{(i)}_{LCS}$.
- Calculation of the optimal REE trajectory to ensure the high-quality of fiber winding on the composite frame. A differential evolution algorithm (see Section 4) is used for the optimization procedure. Determining the optimal sequence $TCP{(i)}_{opt}$ ($1\le i\le N$) is the result of calculation.
- Storing the calculated sequence of $TCP{(i)}_{opt}$ ($1\le i\le N$) in the central robot unit. Determining the optimal trajectory by linking individual corresponding parameters of consecutive following $TCP{(i)}_{opt}$ (using programming instruction of robot—linear interpolations or cubic splines).

**Note 4**

#### 2.3.2. Use of Differential Evolution Algorithm to REE Trajectory Optimization

#### 2.3.3. Pseudo-Code of MDEA

Algorithm 1. MDEA |

Input: The number of calculated generations $NG$, crossover probability $CR$, mutation factor $f$, generation size $NP$, the dimension of individuals $D=4$, lower limits $L{o}_{j}$ and upper limits $H{g}_{j}$, $1\le j\le 4$. Internal computation: - Create an initial generation ($k=0$) of $NP$ individuals ${y}_{m}^{k}$, $1\le m\le NP,$ (e.g., by use of relation (13)).
- a) Evaluate all the individuals ${y}_{m}^{k}$ of the k-th generation (calculate $F({y}_{m}^{k})$ for each individual ${y}_{m}^{k}$). b) Store the individuals ${y}_{m}^{k}$ and their evaluations $F({y}_{m}^{k})$ into matrix $B$ (each matrix row contains parameters of individual ${y}_{m}^{k}$ and its evaluation $F({y}_{m}^{k})$, $1\le m\le NP$).
- while $k\le NG$
collision:=true repeat (i) randomly select index ${s}_{m}\in \{1,\text{}2,\text{}\dots \text{},\text{}D\},$ (ii) randomly select indices ${r}_{1},\text{}{r}_{2},\text{}{r}_{3}\in \{1,\text{}\dots \text{},\text{}NP\},$ where ${r}_{l}\ne m$ for $1\le l\le 3$; ${r}_{1}\ne {r}_{2},\text{}{r}_{1}\ne {r}_{3},\text{}{r}_{2}\ne {r}_{3}$; (iii) for $j:=1$ step $1$ to $D$ do if $rand\text{}(0,\text{}1)$ $\le CR$ or $j={s}_{m}$) then ${y}_{m,j}^{trial}:={y}_{{r}_{3},j}^{k}+f\text{}\left({y}_{{r}_{1},j}^{k}-{y}_{{r}_{2},j}^{k}\right)$ else ${y}_{m,j}^{trial}:={y}_{m,j}^{k}$ end for (j) (iv) Testing of possible collisions of the frame location in $BCS$ defined by ${y}_{m}^{trial}$ and the fiber-processing head. if ${y}_{m}^{trial}$ does not include collisions then collision = false until collision = false end repeat (v) if $F\left({y}_{m}^{trial}\right)\le F\left({y}_{m}^{k}\right)$ then ${y}_{m}^{k+1}:={y}_{m}^{trial}$ else ${y}_{m}^{k+1}:={y}_{m}^{k}$ end if end for (m) b) Store individuals ${y}_{m}^{k+1}$ and their evolutions $F\left({y}_{m}^{k+1}\right)$ $\left(1\le m\le NP\right)$ of the new generation $(k+1)$-st generation in the matrix $B$, $k:=k+1$. c) Find index $p$ which satisfies the condition $F\left({y}_{p}^{k+1}\right)\ge F({y}_{m}^{k})$ for $1\le m\le NP$, ${y}_{p}^{k}:={y}^{rand}$, where ${y}^{rand}$ satisfies (13) end while (k). Output:The best found individual ${y}_{opt}$ is represented by the row of matrix $B$ that contains the corresponding value $\mathrm{min}\{F\left({y}_{m}^{k}\right)\text{};\text{}{y}_{m}^{k}\in B\}$. Comments.The repeat until condition cycle is always executed at least once, since the controlling condition is checked at the end of the cycle. Function $rand\text{}(0,\text{}1)$ randomly picks a number from the interval $\langle 0,\text{}1\rangle $. The notation ${y}_{m,j}^{k}$ means the j-th component of an individual ${y}_{m}^{k}$ in the k-th generation. The individual ${y}_{opt}$ in pseudo-code of MDEA is the final solution and corresponds to designation $y{(i)}_{opt}$ that includes optimized parameters ${x}_{H{(i)}_{opt}},\text{}{z}_{{H(i)}_{opt}},\text{}\phi {(i)}_{opt}$, $\omega {(i)}_{optn}$. However, it should be noted that in general parameters ${x}_{H{(i)}_{opt}},\text{}{z}_{{H(i)}_{opt}},\text{}\phi {(i)}_{opt}$, $\omega {(i)}_{opt}$ calculated by MDEA can only be optimized (and not optimal) parameters in relation to equation (9). This is due to the calculation of the final number of generations of individuals using MDEA. Therefore we mark calculated parameters as optimal parameters ${x}_{H{(i)}_{opt}},\text{}{z}_{{H(i)}_{opt}},\text{}\phi {(i)}_{opt},\text{}\omega {(i)}_{opt}$. |

## 3. Mechanical Performance of Polymer Composite Frame

## 4. Practical Experimental Verification Tests of Optimization Procedure

#### 4.1. Experimental Test 1—Composite Non-Bearing Core Frame Shaped in 2D

#### 4.2. Experimental Test 2—3D Shape Non-Bearing Core Frame

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Example of 3D geometrically complicated non-bearing core frame used to reinforce chassis of passenger car.

**Figure 2.**(

**a**) production process of long-fiber reinforced polymer composite frame, (

**b**) a laboratory mold for holding and curing of the polymer composite frame (example of wind turbine blade), (

**c**) the core before fiber winding, and (

**d**) wound core with fibers, positioned in the mold before injection of the polymer matrix.

**Figure 3.**Right-handed helix ${p}_{R}$ with initial point ${A}_{R}$ and left-handed helix ${p}_{L}$ with initial point ${A}_{L}$.

**Figure 4.**Scheme—front view of the non-bearing core frame with (

**left**) positive and (

**right**) negative angle of fiber winding on the frame. Winding angle $\omega $ is angle between central axis $z$ of the frame and plane of fiber winding ${\chi}_{1}$ (respectively ${\chi}_{2}$).

**Figure 5.**Initial points of the winding right-handed helices ${p}_{R}{}_{i}$ (

**left**) and with rotation angle ψ (

**right**) of the winding layer.

**Figure 6.**Different views of robot KR 16-2 with non-bearing core frame and fiber-processing head with three guide lines.

**Figure 7.**Fiber-processing head in mathematical model in $BCS$ (

**left**) and coordinate systems $BCS$ and $LCS$ (

**right**).

**Figure 8.**Example of a vertical cross-section of a non-bearing core frame with axis $o$ in $LCS$—frame for a baby carriage. The frame is connected to the REE at the point $B{(N)}_{LCS}$, where $N=2131$. Unit tangent vector $b1{(i)}_{LCS}$ to axis ${o}_{LCS}$ and unit vector $b2{(i)}_{LCS}$ are defined at point $B{(i)}_{LCS}$ for $i=\text{}1,\text{}\dots \text{},N.$ All the time $b1{(i)}_{LCS}\perp b2{(i)}_{LCS}$ holds (vector $b2{(i)}_{LCS}$ characterizes the needed rotation of the frame around axis ${o}_{LCS}$ when point $B{(i)}_{LCS}\in {o}_{LCS}$ passes through fiber-processing head).

**Figure 9.**Schematic front view of winding fiber layers onto the non-bearing core frame, case when axis ${S}_{BCS}$ of the fiber-processing head and axis ${O}_{BCS}$ of the frame are identical in displayed section, axis ${S}_{BCS}$ is parallel to the coordinate axis ${y}_{BCS}$.

**Figure 10.**Front view of general scheme of non-bearing core frame passage through the fiber-processing head, axis ${s}_{BCS}$ of fiber-processing head is parallel to axis ${y}_{BCS}$.

**Figure 13.**Schematic front view of collision test of non-bearing core frame passage through fiber-processing head (axis ${s}_{BCS}$ is parallel to axis ${y}_{BCS}$).

**Figure 14.**Image of the polymer composite frames wound using (

**a**) the new and (

**b**) manual-robot winding processes.

**Figure 16.**Test 1—Non-bearing core frame connected to end-effector of industrial robot KUKA KR 16-2 (

**left**). Testing of passage of the frame through fiber-processing head (

**right**).

**Figure 18.**Test 1—the graphical simulation of the robot position and the frame in $BCS$ in the winding at four selected points (

**1**–

**4**) of optimal trajectory.

**Figure 19.**Test 1—Diagram of the optimal course of the $TC{P}_{OPT}$ during the passage of the frame through the fiber-processing head-values of the first three parameters of optimal $TC{P}_{OPT}$.

**Figure 20.**Test 1—Diagram of the optimal course of the $TC{P}_{OPT}$ during the passage of the frame through the fiber-processing head - values of the last three parameters of optimal $TC{P}_{OPT}$.

**Figure 21.**Test 1—Diagram showing the course of function $F(x{(i)}_{opt},\text{}z{(i)}_{opt},\text{}\phi {(i)}_{opt},\text{}\omega {(i)}_{opt})$ for optimal REE trajectory and values $F({x}_{{H}_{BCS}},\text{}{z}_{{H}_{BCS}},\text{}0,\text{}0)$ for non-optimal REE trajectory during the passage of the frame through the fiber-processing head ($1\le i\le N=2131$).

**Figure 22.**Test 2—Frame passage testing through fiber-processing head for calculated optimal REE trajectory.

**Figure 24.**Test 2—Diagram of the optimal course of the $TC{P}_{OPT}$ during the passage of the frame through the fiber-processing head-values of the first three parameters of $TC{P}_{OPT}$.

**Figure 25.**Test 2—Diagram of the optimal course of the $TC{P}_{opt}$ during the passage of the frame through the fiber-processing head—values of the last three parameters of $TC{P}_{opt}$.

**Figure 26.**Experimental Test 2—Diagram showing the course of function $F(x{(i)}_{opt},\text{}z{(i)}_{opt},\text{}\phi {(i)}_{opt},\text{}\omega {(i)}_{opt})$ for optimal REE trajectory and values $F({x}_{{H}_{BCS}},\text{}{z}_{{H}_{BCS}},\text{}0,\text{}0)$ for non-optimal REE trajectory during the passage of the frame through the fiber-processing head ($1\le i\le N=1031$).

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## Share and Cite

**MDPI and ACS Style**

Mlýnek, J.; Petrů, M.; Martinec, T.; Rahimian Koloor, S.S.
Fabrication of High-Quality Polymer Composite Frame by a New Method of Fiber Winding Process. *Polymers* **2020**, *12*, 1037.
https://doi.org/10.3390/polym12051037

**AMA Style**

Mlýnek J, Petrů M, Martinec T, Rahimian Koloor SS.
Fabrication of High-Quality Polymer Composite Frame by a New Method of Fiber Winding Process. *Polymers*. 2020; 12(5):1037.
https://doi.org/10.3390/polym12051037

**Chicago/Turabian Style**

Mlýnek, Jaroslav, Michal Petrů, Tomáš Martinec, and Seyed Saeid Rahimian Koloor.
2020. "Fabrication of High-Quality Polymer Composite Frame by a New Method of Fiber Winding Process" *Polymers* 12, no. 5: 1037.
https://doi.org/10.3390/polym12051037